Z4 -KERDOCK CODES, ORTHOGONAL SPREADS, AND
EXTREMAL EUCLIDEAN LINE-SETS
A. R. CALDERBANK, P. J. CAMERON,
W. M. KANTOR, and J. J. SEIDEL
[Received 27 November 1995—Revised 25 July 1996]
1. Introduction
Kerdock and Preparata codes are non-linear binary codes that contain more
codewords than any comparable linear codes presently known; see [6, 20, 29, 28].
These codes have excellent error-correcting capabilities and have the remarkable
property of being ‘formal duals’, in the sense that although these codes are
non-linear, the distance distribution of either is the MacWilliams transform of the
distance distribution of the other (see MacWilliams and Sloane [28, Chapter 15]).
Kerdock and Preparata codes exist for all lengths 2m+1 > 16, where throughout
this introduction m will denote an odd integer greater than or equal to 3. If
2m+1 > 16, these codes are not unique in the sense that several codes exist with
the same weight distributions [3, 16, 17, 18, 26].
Recently, Hammons, Kumar, Calderbank, Sloane and Solé [14] showed that the
Kerdock and Preparata codes can be very simply constructed as binary images
under a certain natural map, called the Gray map, of linear codes over Z4
(although this requires a slight modification of the notion of ‘Preparata’ codes:
the new ones and the classical ones have the same weight distribution but they
are not isomorphic if 2m+1 > 16). The Gray map will be defined and studied later
in x 8. For now we note its main property: it is an isometry (Z4N , Lee metric) →
(Z22 N , Hamming metric), where the Lee metric is a standard metric defined in x 8.
One of the main results of the present paper is an explanation of the Gray
map in terms of suitable finite groups and geometries (Theorem 8.3). En route
to this, we will see connections between the geometry of binary orthogonal and
symplectic vector spaces on the one hand, and extremal real and complex line-sets
having only two angles on the other. Extraspecial groups will be the source of
these connections.
This research can be viewed as beginning in a paper by Cameron and Seidel
[9]. That paper used quadratic forms on Z2m+1 in order to construct a family of
1 2
N
m +1
2 N lines through the origin of R , where N = 2√ , such that any two are either
perpendicular or at an angle θ, where cos θ = 1/ N . In fact, their line-sets were
unions of 12 N frames (that is, N pairwise orthogonal 1-spaces); these sets appear
in the right-hand portion of Corollary 3.5 (but in notation quite different from that
of [9]). Recently, König [22] observed that this line-set is not extremal (also see
Levenštein [25]). While constructing the isometric embedding `2N → `4N ( N +1)/2 (cf.
Research of W. M. Kantor was supported in part by NSF and NSA grants.
1991 Mathematics Subject Classification: primary 94B60; secondary 51M15, 20C99.
Proc. London Math. Soc. (3) 75 (1997) 436–480.
Z4 - KERDOCK CODES
437
Lyubich and Vaserstein [27]) he augmented the set of 12 N 2 lines from [9] by the
standard coordinate frame. This did not increase the set of prescribed angles (this
observation also is tucked into Corollary 3.5). The augmented system of lines
meets an upper bound obtained by Delsarte, Goethals and Seidel [11]. Whereas
their proof used Jacobi polynomials, we will give a proof of this upper bound
in x 2 using elementary tensor algebra; we will also discuss the graph-theoretic
structure of any system of lines meeting the upper bound.
The construction in [9] used binary Kerdock codes. Therefore, it was natural to
imitate that construction by using the Z4 -linear Kerdock code of length N 0 = 2m
0
discovered by Hammons et al. [14]. This results in a set of√N 0 2 lines in C N that
are either perpendicular or at an angle θ0 , where cos θ0 = 1/ N 0 . Once again, this
set is a union of pairwise disjoint frames (this set appears later in the right-hand
portion of Corollary 5.7). Adding the coordinate frame extends this to a line-set
0
in C N of size N 02 + N 0 which has these angles but meets another upper bound;
this result is due to Delsarte, Goethals and Seidel [11].
The extremal real and complex line-sets in the preceding two paragraphs were
constructed by adding the standard coordinate frame to a line-set arising from a
version of a Kerdock code. This construction seemed to us too ‘inhomogeneous’,
in the sense that the line-set was a union of frames, but one frame was somehow
distinguished. This raised the question of whether there might be a more uniform
description of all of the frames, a description that did not involve singling out
any one frame. This question also suggested a direction in which to look for
an answer: a similar situation had been encountered in other research concerning
binary Kerdock codes.
Kantor [16, I] investigated connections between orthogonal spreads and Kerdock
sets. When the vector space Z22m+2 is equipped with the quadratic form
x1 xm+2 + x2 xm+3 + . . . + xm+1 x2m+2 , there are (2m+1 − 1 )(2m + 1 ) singular points
(that is, 1-spaces on which the form vanishes), comprising a hyperbolic quadric
in the corresponding projective space. An orthogonal spread is a partition of this
set of points into 2m + 1 totally singular (m + 1 )-spaces. In [16, I], orthogonal
spreads are used to construct Kerdock codes and vice versa. We will review
this correspondence in x 3. As in the preceding paragraph, the construction also
distinguishes one (arbitrary) member of the spread as part of the relationship with
Kerdock codes (see the remark following Corollary 3.5 below). This was too
much of a coincidence, suggesting that there had to be some way to pass between
the binary world of orthogonal spreads and the real or complex world of frames
and line-sets.
Most of this paper is concerned with the bridge between these differentlyflavoured geometries: extraspecial 2-groups. Section 2 describes the groups we
require, as well as their unique faithful irreducible real representations. There
is nothing new here from a group-theoretic point of view. Each such group E
has order 21+2k for some integer k , and arises as a group of isometries of the
k
Euclidean space R2 . On the other hand, E / Z ( E ) naturally inherits a quadratic
form, producing the desired binary geometry. Section 3 describes how these two
geometries associated with E produce extremal real line-sets when k = m + 1 and
m is odd as before.
In xx 4 and 5 we change groups slightly in order to connect symplectic spreads,
Z4 -Kerdock codes and extremal complex line-sets. In x 6 we describe how to
438
A . R . CALDERBANK
et al.
m+1
pass from extremal line-sets in R2
derived from orthogonal spreads to extremal
m
line-sets in C2 derived from symplectic spreads.
Section 7 gives a simple geometric construction of an orthogonal spread
in E / Z ( E ) starting with a symplectic spread (taken from Kantor [16, I]). A
geometric correspondence between symplectic and orthogonal spreads is also
expressed in computational terms as a correspondence between binary symmetric
m × m matrices and binary skew-symmetric (m + 1 ) × (m + 1 ) matrices.
Symplectic spreads in a binary vector space determine Z4 -Kerdock codes, while
orthogonal spreads determine binary Kerdock codes. In Theorem 8.3 we show that
the geometric/group-theoretic map from symplectic spreads to orthogonal spreads
induces the Gray map from a corresponding Kerdock code over Z4 to its binary
image.
Examples of Kerdock and Preparata codes are given in x 9. Code equivalence
is discussed at length in x 10. Large numbers of inequivalent Z4 -Kerdock codes
are constructed (using results of Kantor [18]). We will see that the construction
of new Z4 -linear Preparata codes amounts to the construction of certain types of
affine translation planes and certain types of division algebras (Proposition 8.9 and
Corollary 8.11). Using this we will obtain Z4 -linear Preparata codes not equivalent
to those constructed by Hammons et al. [14].
Finally, in x 11 we consider the analogue of the binary case using extraspecial
p-groups for odd primes p, in order to relate symplectic spreads and extremal
m
line-sets in C p . However, the behaviour of extraspecial groups when p is odd
is known to be quite different from that when p = 2. In particular, orthogonal
geometry does not enter at all, nor does any analogue of Kerdock codes.
For the convenience of the reader we have included a ‘road map’ of the paper
in Table 1. Numbers in parentheses indicate Sections.
2. Extraspecial 2-groups
Let k be a fixed positive integer and let N = 2k . In later sections we will
take k = m + 1 when considering orthogonal spreads and k = m when considering
symplectic spreads.
For a prime p, a p-group P is said to be extraspecial if the centre Z ( P ) has
order p and if P/ Z ( P ) is elementary abelian (and hence a vector space over
GF( p )).
We begin with a description of an extraspecial 2-group E = E k of order 21+2k as
an irreducible group of orthogonal N × N matrices with real entries. Since E has
22k distinct linear characters and 21+2k = 22k · 12 + (2k )2 , this will be the unique
faithful irreducible representation of E . We also construct explicitly a group L of
real orthogonal transformations containing E as a normal subgroup. Elements of
L act on E by conjugation, fixing the centre Z ( E ) of order 2. Hence there is a
well-defined action on the elementary abelian group E = E / Z ( E ) of order 22k .
We will see that this action on E preserves an explicit non-singular quadratic form
Q, providing a bridge between binary orthogonal and real orthogonal geometry.
Let V denote the vector space Z2k . For x, y ∈ V , let x · y denote the usual dot
product.
Equip R N with the usual inner product, and let O (R N ) denote the corresponding
group of orthogonal linear transformations. We will tend to identify orthogonal
matrices with the corresponding orthogonal transformations.
439
Z4 - KERDOCK CODES
TABLE 1. A map of xx 2–10
Discrete Z2 -world (3)
Real world (2)
m+1
O (R2
L
E
E = E / Z (E )
elementary abelian
of order 22(m+1)
induced action on E
preserving the
quadratic form Q (e ) = e2
(2 ), (3 )
←−−−−−−−−−→
orthogonal spread 6:
partition of the
(3 )
(2m+1 − 1 )(2m + 1 )
−−−−−−−−−−−→
singular points by 2m + 1
totally singular (m + 1 )-spaces



(6 ) 

y
induced action on F
preserving the alternating
form ( f 1 , f 2 ) = [ f 1 , f 2 ]
L < N O (R2m+1 ) ( E )
L/ E ∼
= O+ (2m + 2, 2 )
extraspecial 2-group
of order 21+2(m+1)
L acts on E by conjugation
collection F(6) of 2m + 1
orthogonal frames each
containing 2m+1 1-spaces



 Gray Map (8 ), (9 ), (10 )

y



 (7 )

y
symplectic spread 60 :
partition of the
(2m − 1 )(2m + 1 )
points by 2m + 1
totally isotropic m-spaces
)
(5 )
−−−−−−−−−−−→
collection F(60 ) of 2m + 1
unitary frames each containing
2m complex 1-spaces
(4 ), (5 )
←−−−−−−−−−→
L\ acts on F by conjugation
m
F = F/ Z ( F )
elementary abelian
of order 22m
Discrete Z4 -world (5)
U (C 2 )
L\
F = E \ hiI i
L\ < NU (C2m ) ( F )
L\ / F ∼
= Sp(2m, 2 )
2-group
of order 22+2m
Complex world (4)
Label the standard basis of R N as ev , with v ∈ V . For b ∈ V , define the
permutation matrix X (b ) and diagonal matrix Y (b ) as follows:
X (b ) : ev → ev+b
and Y (b ) := diag[ (−1 )b·v ].
The groups X (V ) := { X (b ) | b ∈ V } and Y (V ) := {Y (b ) | b ∈ V } are contained
in O (R N ) and are isomorphic to the additive group V . Let E := h X (V ), Y (V )i;
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A . R . CALDERBANK
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this is an irreducible subgroup of O (R N ). We will spend most of this paper
examining E , in spite of the fact that this group, and this representation, have
been thoroughly investigated within group-theoretic contexts [15, pp. 355–357; 1,
p. 109; 32, pp. 97–98; 21, pp. 148–155].
L EMMA 2.1. The group E = h X (V ), Y (V )i is extraspecial of order 21+2k , and
Z ( E ) = h− I i. Moreover, E = X (V )Y (V )h− I i, and every element of E can be
written uniquely in the form X (a )Y (b )(− I )γ for some a, b ∈ V and γ ∈ Z2 .
Proof. For all a, b ∈ V ,
X (a )−1 Y (b )−1 X (a )Y (b ) = (−1 )a·b I
(2.2)
since
ev ( X (a )−1 Y (b )−1 X (a )Y (b )) = [ (−1 )b·(v+a ) ev+a ] ( X (a )Y (b ))
= (−1 )b·(v+a ) (−1 )b·v ev .
It follows that multiplication in E is given by the formula
0
( X (a )Y (b )(− I )γ )( X (a0 )Y (b0 )(− I )γ ) = X (a )(Y (b ) X (a0 ))Y (b0 )(− I )γ +γ
0
0
0
= X (a + a0 )Y (b + b0 )(− I )γ +γ +a ·b (2.3)
for all a, b, a0 , b0 ∈ V and γ, γ 0 ∈ Z2 . Hence E = X (V )Y (V )h− I i, Z ( E ) = h− I i,
and E / Z ( E ) ∼
= V ⊕ V . The uniqueness assertion is clear.
The natural map from E to E = E / Z ( E ) will be used very frequently; an
overbar will signify an image under this map (for example, the image of X (a )
will be denoted X (a )).
We identify the centre Z ( E ) of E with Z2 and consider the map Q : E → Z2
defined by Q (e ) = e2 for any e ∈ E and any preimage e of e in E . Then Q is a
non-singular quadratic form on E . The associated alternating bilinear form on E
is given by (e 1 , e 2 ) E = [e1 , e2 ], where [ x, y] = x−1 y−1 xy denotes the commutator
of x and y; we will generally drop the subscript ‘ E ’ when there is no possibility
of confusion with the inner product on R N . From (2.3) and (2.2) we have
Q ( X (a )Y (b )) = a · b,
(2.4)
( X (a )Y (b ), X (a0 )Y (b0 )) = a · b0 − a0 · b.
(2.5)
Note that the k -spaces X (V ) and Y (V ) are totally singular: the quadratic form Q
vanishes on each of them. Also, X (V ) ∩ Y (V ) = 0. Hence, E is an + (2k , 2 )space: it has maximal Witt index (or, equivalently, the associated quadric is
hyperbolic). For more information about extraspecial groups and quadratic forms
we refer the reader to various texts [15, pp. 355–357; 1, p. 109; 32, pp. 97–98;
33].
We will require isometries of R N that normalize E . These isometries normalize
the centre Z ( E ), act on E by conjugation, and induce isometries of E . Of
particular interest is the matrix
1
H = √ [ (−1 )u·v ]u,v∈V .
N
(2.6)
441
1 1
Z4 - KERDOCK CODES
√
Here N H is the k -fold Kronecker product of the 2 × 2 Hadamard matrix 1 −1
(so that H 2 = I ) and is also the character table of V (the equation H 2 = I
expresses orthogonality of characters). We have
1 X
a·v
(−1 ) ev+b H
ea H X (b ) H = √
N v
1X
(−1 )a·v+(v+b )·w ew
=
N v,w
X
1X
(−1 )a·b
(−1 )(a+w)·(v+b ) ew
=
N w
v
= (−1 )a·b ea ,
from which it follows that
Define the unit vector
H −1 X (b ) H = H X (b ) H = Y (b ).
(2.7)
1 X
(−1 )b·v ev
eb∗ := √
N v
(2.8)
to be the ‘bth’ row of H .
In addition to the standard basis of R N we will need to choose a basis of
E. Let v1 , . . . , vk denote the standard basis of V = Zk2 . This determines bases
x1 , . . . , xk of X (V ) and y1 , . . . , yk of Y (V ), namely x j = X (v j ) and y j = Y (v j ).
These are essentially ‘dual’ bases of X (V ) and Y (V ) with respect to ( , ) E :
(xi , y j ) = δij for all i, j. We will always write matrices of linear transformations
on E with respect to the ordered basis x1 , . . . , xk , y1 , . . . , yk . By (2.7), H induces
the map x j ↔ y j on E .
Every matrix A in the general linear group GL(V ) induces an orthogonal
transformation A˜ of R N by permuting coordinates: ev 7→ ev A . We will identify
GL(V ) with { A˜ | A ∈ GL(V )}.
˜
L EMMA
2.9. (i) The orthogonal transformation A normalizes E and induces
on E by conjugation:
A O
.
A˜ −1 X (a )Y (b ) A˜ = X (a A )Y (b A −T ) = X (a )Y (b )
O A −T
A O
O A− T
(ii) The group GL(V ) acts transitively by conjugation on both
{ X (a )Y (b ) | a · b = 0, a , b 6= 0} and
{ X (a )Y (b ) | a · b = 1}.
Proof. (i) For all b, v ∈ V we have
ev ( A˜ −1 X (b ) A˜ ) = ev A−1 +b A˜ = ev+b A = ev X (b A ) ,
so that A˜ −1 X (b ) A˜ = X (b A ). Furthermore,
−1
−T
ev ( A˜ −1 Y (b ) A˜ ) = ev A−1 (−1 )v A ·b A˜ = (−1 )v·b A ev = ev Y (b A −T ) ,
so that A˜ −1 Y (b ) A˜ = Y (b A −T ). This implies (i).
(ii) The group GL(V ) acts transitively on the set of incident point-hyperplane
pairs from V as well as on the set of non-incident point-hyperplane pairs from V .
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A . R . CALDERBANK
et al.
Later we will need still more isometries of both E and R N . It is straightforward
to check that the isometries of E that induce the identity on Y (V ) are precisely
those described by matrices OI MI , where M is a skew-symmetric k × k matrix
(recall that a skew-symmetric matrix has zero diagonal, by definition). We require
an isometry d M of R N that induces OI MI on E by conjugation.
To this end let Q M be any quadratic form on V for which the associated
bilinear form is uM v T , so that Q M (u + v) = Q M (u ) + Q M (v) + uM v T for all
u , v ∈ V . Let
(2.10)
d M := diag[ (−1 ) Q M (v) ].
Then
1
Q M (v)
ev d −
ev+a d M X (a )
M X (a )d M X (a ) = (−1 )
Q M (v)+ Q M (v+a )
ev
= (−1 )
Q M (a )
(aM )·v
ev ,
(−1 )
= (−1 )
Q M (a )
1
so that d −
Y (aM ). Hence,
M X (a )d M X (a ) = (−1 )
1
d−
M ( X (a )Y (b ))d M
= X (a )Y (aM )Y (b ) = X (a )Y (b )
I
O
M
I
.
(2.11)
Note that the diagonal matrix d M depends on the choice of the quadratic form
Q M , but that the effect of conjugation by d M is independent of this choice.
This proves part of the next lemma. The remainder of the lemma further relates
totally singular k -spaces of E to the corresponding skew-symmetric matrices; the
proof is straightforward (see Kantor [16, I, x 5]).
L EMMA 2.12. (i) Every totally singular k-space W of E such that Y (V ) ∩ W = 0
has the form
I M
−1
W = d M X (V )d M = X (V )
= { X (a )Y (aM ) | a ∈ V }
O I
for a unique binary skew-symmetric
k × k matrix M. The linear transformation of
E produced by OI MI preserves the quadratic form Q.
(ii) Let M1 and M2 be binary skew-symmetric k × k matrices for which
the corresponding totally singular k-spaces W1 and W2 satisfy Y (V ) ∩ W1 =
Y (V ) ∩ W2 = 0. Then W1 ∩ W2 = 0 if and only if M1 − M2 is non-singular.
We need one further element of O (R N ) that normalizes E : one that induces a
transvection on E . First note that
E = E k = X (hv1 i )Y (hv1 i )h− I i · X (hv2 , . . . , vk i )Y (hv2 , . . . , vk i )h− I i
can be viewed as E 1 ⊗ E k −1, the tensor (or Kronecker)
product of groups of
2 × 2 and 2k −1 × 2k −1 matrices. Let H2 = √12 11 −11 and H˜ 2 = H2 ⊗ I . Then H2
normalizes the dihedral group h X (v1 ), Y (v1 )i = X (hv1 i )Y (hv1 i )h− I i of order 8,
interchanging X (v1 ) and Y (v1 ) and fixing h X (v1 )Y (v1 )i. (All of this is the
special case k = 1 of (2.7).) It follows that H˜ 2 normalizes E while centralizing
X (hv2 . . . , vk i )Y (hv2 . . . , vk i )h− I i. Thus, H˜ 2 induces a transvection on E.
We now have enough isometries of E to generate the orthogonal group
O+ (2k , 2 ) using conjugation by isometries in O (R N ) that normalize E (this is
Z4 - KERDOCK CODES
443
not the group + (2k , 2 ), which has index 2 in O+ (2k , 2 ) and is simple if k > 3).
Namely, if
L := h E , GL(V ), d M , H , H˜ 2 | M is skew-symmetrici,
(2.13)
then L contains E as a normal subgroup and L / E acts on E by conjugation.
Note that L contains all isometries of E that fix Y (V ), and hence all that
fix H −1 Y (V ) H = X (V ); these two groups of isometries generate + (2k , 2 ) [1,
(43.7)]. Since H˜ 2 induces a transvection on E , it follows that
L EMMA 2.14. The group L induces O+ (2k , 2 ) on E.
3. Binary Kerdock codes, orthogonal spreads, and real line-sets
with prescribed angles
Fix an odd integer m, and let k = m + 1 in the preceding section. Now E = E m+1
is an extraspecial 2-group of order 21+2(m+1) , and E is an + (2m + 2, 2 )-space.
D EFINITION . An orthogonal spread of E is a family 6 of 2m + 1 totally
singular (m + 1 )-spaces such that every singular point of E belongs to exactly
one member of 6. (Note that E contains (2m+1 − 1 )(2m + 1 ) singular points,
that is, singular 1-spaces, of which 2m+1 − 1 are in any given totally singular
(m + 1 )-space.)
The construction of orthogonal spreads will be discussed in xx 7 and 9. In this
section we will describe their relationship to Kerdock codes.
By Lemma 2.14, L acts transitively on the ordered pairs of disjoint (except for
0) totally singular (m + 1 )-spaces of E . After replacing 6 by 6` for some ` ∈ L we
may assume that X (V ), Y (V ) ∈ 6. By Lemma 2.12(i), any A ∈ 6 \ {Y (V )} can be
written in the form X (V ) OI MI A for a unique skew-symmetric (m + 1 ) × (m + 1 )
matrix M A . The matrix OI MI A (or, to be more precise, the linear transformation
it induces) preserves the quadratic form Q defined on E (cf. (2.4)).
Consider a set of binary skew-symmetric (m + 1 ) × (m + 1 ) matrices such that
the difference of any two is non-singular. Since the first rows of matrices in this
set must be distinct, such a set has size at most 2m . The important case is when
this bound is achieved.
D EFINITION . A Kerdock set is a set of 2m binary skew-symmetric (m + 1 ) ×
(m + 1 ) matrices such that the difference of any two is non-singular.
By Lemma 2.12(ii), { M A | A ∈ 6 \ {Y (V )}} is a Kerdock set for any orthogonal
spread 6 containing Y (V ). The corresponding Kerdock code K(6) is a binary
code of length 2m+1; its 2m+1 coordinate positions are labelled by vectors in
V = Z2m+1 , and
K(6) := {( Q M A (v) + s · v + ε)v∈V | A ∈ 6 \ {Y (V )}, s ∈ V, ε ∈ Z2 },
(3.1)
where, for each A ∈ 6 \ {Y (V )}, Q M A denotes any quadratic form associated with
the alternating bilinear form uM A v T . Thus, |K(6)| = 2m · 2m+1 · 2 = 22m+2 .
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A . R . CALDERBANK
et al.
R EMARK . It is important to observe that this notation is very ambiguous. The
Kerdock set depends on the choice of a pair of members of 6 to play the roles of
X (V ) and Y (V ), and on the choice of dual bases in X (V ) and Y (V ). However,
up to equivalence of codes, the Kerdock code K(6) depends only on 6 and on
the distinguished member Y (V ) of 6 that was discarded in the construction of
the Kerdock set. In choosing the notation K(6) rather than, say, K(6, Y (V )), we
have suppressed this dependence. This is just the sort of inhomogeneity alluded
to in x 1. See Kantor [16] for more information about this construction and
the unavoidability of this dependence. Expositions of this relationship between
orthogonal spreads and Kerdock codes are given in [26] and [8, pp. 143–148].
(Note. On p. 147 of [8] there is an incorrect assertion that there is a one-to-one
correspondence between Kerdock sets and orthogonal spreads.)
Next we consider the Euclidean geometry of the extraspecial 2-group E .
L EMMA 3.2. If eb∗ is as in (2.8) , then
(i) {hev i | v ∈ V } is the set of irreducible submodules for Y (V );
(ii) {heb∗ i | b ∈ V } is the set of irreducible submodules for X (V ).
Proof. (i) Since Y (V ) is a group of diagonal matrices it certainly leaves
invariant each of the 1-spaces hev i. If u, v ∈ V are distinct, then there exists b ∈ V
with b · u 6= b · v, and then Y (b ) acts differently on heu i and hev i. Thus, we have
2m+1 inequivalent 1-dimensional submodules of an 2m+1 -dimensional module, so
these comprise all irreducible submodules.
(ii) By (2.7), the 1-spaces heb H i = heb∗ i are the irreducible submodules of
−1
H Y (V ) H = X (V ).
L EMMA 3.3. Let A and B be subgroups of E such that A and B are totally
singular (m + 1 )-spaces of E.
m+1
(i) The set F( A ) of A-irreducible subspaces of R2
is an orthogonal frame:
a set of 2m+1 pairwise orthogonal lines through the origin.
(ii) Assume that A ∩ B = 0, and let u1 and u2 be unit vectors in different
members of F( A ) ∪ F( B ). If u1 and u2 are both in different members of
F( A ), or both are in different members of F( B ), then (u1 , u2 ) = 0; otherwise,
|(u1 , u2 )| = 2−(m+1)/2 .
(iii) The set F( A ) is left invariant by E.
Proof. In view of the transitivity of L on the ordered pairs A, B in (ii), we
may assume that A = X (V ) and B = Y (V ). Part (i) now follows directly from
Lemma 3.2(i). Also by Lemma 3.2 and (2.8), (ea , eb∗ ) = 2−(m+1)/2 (−1 )a·b for all
a, b ∈ V ; since (u1 , u2 ) = 0 whenever hu1 i and hu2 i belong to the same orthogonal
frame, this proves (ii). To prove part (iii) we observe that h A , − I i is a normal
subgroup of E , so that E leaves invariant the set F( A ) of irreducible submodules
for h A , − I i.
445
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R EMARK . The issue of whether or not a subgroup A of E that projects onto
A contains − I is immaterial in the above lemma, and will be immaterial in the
sequel.
T HEOREM 3.4. Let 6 be an orthogonal spread of the + (2m + 2, 2 )-space E,
and let
[
F(6) :=
F( A ).
A∈6
m+1
Then F(6) consists of 2m+1 (2m + 1 ) lines of R2
such that, if u 1 and u 2 are
unit vectors in different members of F(6), then |(u1 , u2 )| = 0 or 2−(m+1)/2 .
Proof. This follows immediately from Lemma 3.3.
The Kerdock code K(6) introduced in (3.1) can be recovered very simply from
F(6):
m+1
C OROLLARY 3.5. K(6) = {(cv )v ∈ Z22
| h((−1 )cv )v i ∈ F(6)}.
Proof. By (2.8) and (2.10), F(6)\{F(Y ( V ))} consists of 1-spaces each of
which contains exactly two vectors of length 2(m+1)/2 , namely ±2(m+1)/2 e∗b d M =
P
± v (−1 )b·v (−1 ) Q M (v) ev with M = M A for A ∈ 6\{Y (V )}. Now use (3.1).
R EMARK . The sets F(6) are essentially not new: they arose in work of
Cameron and Seidel [9], König [22] and Levenštein [25]. Namely, while
investigating combinatorial properties of the sets of quadratic and alternating
bilinear forms on an even-dimensional binary vector space, Cameron and
Seidel constructed what, in the present notation, amounts to the line-sets
F(6)\{F(Y (V ))}. Later, König and Levenštein independently observed that one
could add the frame determined by the standard basis vectors and maintain the
same inner products.
Bounds for line-sets in R N with prescribed angles
The line-sets F(6) in Theorem 3.4 are extremal in the sense that |F(6)| meets
an upper bound obtained by Delsarte, Goethals and Seidel [11] for line-sets in R N
with prescribed angles (cf. [4]). We now derive this upper bound using elementary
tensor algebra, and we describe the graph-theoretic structure of extremal line-sets
that meet the bound.
Fix any positive integer N , let e1 , . . . , e N be the standard basis of R N and let
S denote the unit sphere in R N . Consider a spanning set  of n points of S
such that |(a, b )| ∈ {0, α} for all a 6= b in , where 0 < α < 1 (so, in particular,
 ∩ (−) = ∅). Write the Gram matrix of  as
Gram() = I + αC ,
where C is an n × n symmetric matrix with diagonal entries 0 and all other entries
0, ±1. Since  spans R N , Gram() has nullity n − N and the spectrum of C
has the form {(−1/α)n− N , λ1 , . . . , λ N } for some real numbers λi (none of which
is −1/α).
446
A . R . CALDERBANK
et al.
3
N
N
Let
S (R ) denote the space of symmetric 3-tensors of R , of dimension
; for proofs of this and other elementary facts about tensor algebra see [31].
For each a ∈ , S3 (R N ) contains the tensors va := a ⊗ a ⊗ a and
P
ha := 13 iN=1 (a ⊗ ei ⊗ ei + ei ⊗ a ⊗ ei + ei ⊗ ei ⊗ a ).
N +2
3
We begin by showing that
Gram (va ; ha ) := Gram ({va , ha | a ∈ } )
I + α3 C
I + αC
=
.
I + αC 13 ( N + 2 )( I + αC )
(3.6)
Namely, for all a, b ∈ ,
(va , vb ) = (a, b )3 ,
(vb , ha ) = 3 × 13 (a, b )
(ha , hb ) =
3
9
PN
N
X
(b, ei )2 = (a, b ),
i =1
2
i=1 (a , b )(ei , ei )
+ 69
PN
i=1 (a , ei )(b , ei )(ei , ei )
= 13 ( N + 2 )(a, b ),
which proves (3.6).
The n × n matrix I + α3 C is the sum of a positive definite matrix (1 − α2 ) I
and a positive semidefinite matrix α2 Gram() = α2 ( I + αC ), and hence is nonsingular and positive
definite. As seen above, it is also the Gram matrix of n
vectors va in N 3+2 -space. This proves that
N +2
(3.7)
.
n<
3
This inequality is the absolute bound obtained by Delsarte, Goethals and Seidel:
it is independent of the exact angles of the line-set.
In order to go further, simultaneously apply elementary row and column
operations to (3.6) in order to see that the matrix


I + α3 C − N 3+2 ( I + αC )
O


N +2
+
(
α
)
O
I
C
3
is also positive semidefinite. Then so is
( N + 2 )( I + α3 C ) − 3( I + αC ) = ( N + 2 )(1 − α2 ) I − [3 − α2 ( N + 2 )] ( I + αC ).
This implies that, for i = 1, . . . , N ,
0 < 1 + αλi <
( N + 2 )(1 − α2 )
3 − ( N + 2 )α2
assuming that the denominator is positive. Since n = Tr ( I + αC ) =
we deduce that
N ( N + 2 )(1 − α2 )
n<
.
3 − ( N + 2 )α2
(3.8)
PN
This is the special bound obtained by Delsarte, Goethals and Seidel.
i=1 (1 + αλ i ),
(3.9)
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447
If equality holds in (3.9), then all of the 1 + αλi are equal to their upper bound
given in (3.8), and C has two eigenvalues: −1/α with multiplicity n − N , and
.
( N + 2 )(1 − α2 )
n− N
−1
β :=
α=
(3.10)
2
Nα
3 − ( N + 2 )α
with multiplicity N . Consequently,
1
C + I (C − β I ) = 0.
α
(3.11)
However, more can be said in this case (see Cameron and van Lint [8] for the
definition and basic theory of strongly regular graphs):
‘perpendicularity’ defines
P ROPOSITION 3.12. If equality holds in (3.9 ), then √
a strongly regular graph on . In particular, if α = 1/ N then  is a union of
1
2 ( N + 2 ) orthonormal bases, with points in different bases not perpendicular.
Proof. Since (a ⊗ a, b ⊗ b ) = (a, b )2 , we have Gram{a ⊗ a | a ∈ } = ((a, b )2 ) =
I + α2 A , where A is a symmetric (0, 1 ) matrix. Note that the row sums of A are
all β/α. For, each such row sum is a diagonal entry of C 2 , and these are all β/α
by (3.11) since the diagonal entries of C are all 0.
Since dim S2 (R N ) = N 2+1 , the matrix A has at least n − N 2+1 eigenvalues
equal to −1/α2 . Another eigenvalue is β/α, so there are N 2+1 − 1 unknown
eigenvalues ρ j , all of which are real. We have
X
N +1
ρj + n −
(−1/α2 ),
0 = Tr A = β/α +
(3.13)
2
j
nβ/α = Tr A = β /α +
2
2
2
X
j
ρ2j
N +1
+ n−
(1/α4 ).
2
Using (3.10), we find that
X
1 = 12 ( N + 2 )( N − 1 ),
X
ρ j = −( N + 2 )( N − 1 )(1 − N α2 )/2α2 1,
X
ρ2j = ( N + 2 )( N − 1 )(1 − N α2 )2 /2α4 12 ,
where 1 = 3 − ( N + 2 )α2 . Thus, by the Cauchy–Schwarz inequality,
ρ j = −(1 − N α2 )/α2 1
for j = 1, . . . , 12 ( N + 2 )( N − 1 ), and hence the graph is strongly regular.
If α2 = 1/ N then ρi = 0, and the perpendicularity graph is the union of
1
1
2 ( N + 2 ) disjoint N -cliques:  is the union of 2 ( N + 2 ) orthonormal bases, with
members of different bases not perpendicular.
The case α2 = 1/ N is the one appearing in Theorem 3.4. It is natural to ask
whether or not all extremal line-sets in the ‘Kerdock case’ α2 = 1/ N are derived
from orthogonal spreads, but this question appears to be difficult. Suppose that
F is such an extremal line-set. By taking one of its frames as the (standard)
448
A . R . CALDERBANK
et al.
coordinate frame, we can describe F by 12 N ( N + 2 ) unit vectors, one per
line modulo switching. The Gram matrix of these vectors has rank N , and is
partitioned into square blocks of size N : identity matrices on the diagonal and
(normalized) Hadamard matrices elsewhere (cf. [9]). This leads to difficult open
questions concerning the relationships between such extremal line-sets F and
line-sets F(6) originating from orthogonal spreads 6. We do not even know
whether N must have the form 2m+1. If it does, and if F is converted to a binary
code C by changing 1 and −1 to 0 and 1, respectively, then C has the weight
distribution of a Kerdock code. However, we do not know whether the code C
must be equivalent to one arising from some orthogonal spread. Each Hadamard
submatrix of the above Gram matrix produces a subcode of C having the same
weight distribution as a first-order Reed–Muller code, but we cannot prove that
this needs to be an actual Reed–Muller code, nor even that C must lie inside a
second-order Reed–Muller code.
In order to understand this situation somewhat better, consider a set consisting
of d orthonormal bases behaving as in Proposition 3.12. We may assume that
one of the bases is the standard one. Then each of the other bases consists of
the rows of an orthogonal matrix; by assumption, it is a normalized Hadamard
matrix. For d = 2, any Hadamard matrix can occur. However, for d > 2, there
are further restrictions, coming from the fact that the transition matrices Hi−1 H j
between these bases are themselves normalized Hadamard matrices.
For instance, the case d = 3 is equivalent (with a change in notation) to
the existence of three Hadamard matrices H1 , H2 , H3 of order N satisfying
H1 H2 H3 = N 3/2 I . Note that N must be a square. (This also follows from the fact
that the eigenvalue −1/α of the Gram matrix is rational, since its multiplicity is
greater than that of any other eigenvalue.)
Examples can be constructed as follows. Let N = 16s2 , and suppose that there
exists a net of order 4s and index r . (See Cameron and van Lint [8, Chapter 7]
for the definition of a net.) If r > 2s, let P1 be a set of 2s parallel classes, and let
A 1 be the adjacency matrix of the point graph of the resulting sub-net of order
2s. Then H1 = 2 A 1 − J is a symmetric Hadamard matrix. If P2 is another 2s-set
of parallel classes such that | P1 ∩ P2 | = s (this requires that r > 3s), we find
that H1 H2 = −4sH3 , where H3 corresponds to P3 = P1 4 P2 . So, if there exists
a family of d − 1 2s-subsets of an r -set, any two intersecting in s points, then
we obtain d − 1 Hadamard matrices with the corresponding properties, and hence
d orthonormal bases having the properties of Proposition 3.12. Note that this
construction produces far fewer than the potential maximum number 12 ( N + 2 )
of bases. It also shows that sets of bases can be obtained in which inequivalent
Hadamard matrices occur—unlike the case of Lemma 3.3, in which all of the
Hadamard matrices are equivalent to the one produced in Lemma 3.2 (Hadamard
matrices of ‘Sylvester type’).
Much more general results can be found in a paper by Delsarte, Goethals and
Seidel [11]. They obtained very general bounds for finite subsets  of S having
prescribed angles, as well as combinatorial consequences when equality holds.
The geometry of the line-sets F(6)
m+1
L EMMA 3.14. Let G be the group of all elements of O (R2 ) that send each
of the frames F( X (V )) and F(Y (V )) to itself. Then G normalizes E.
Z4 - KERDOCK CODES
449
Proof. Let g ∈ G. Since X (V ) is transitive on the standard basis vectors ev ,
with v ∈ V , we may replace g by its product with an element of X (V ), and
assume that g fixes he0 i ∈ F(Y (V )). In fact we may now replace g by a scalar
multiple of itself and assume that g fixes e0 .
Similarly Y (V ) is transitive on F( X (V )), while inducing the identity on
F(Y (V )). Arguing as above we may also assume that g fixes he∗0 i ∈ F( X (V )).
We will prove that g is in the general linear group GL(V ) occurring in
Lemma 2.9. This will suffice, since by that lemma the group GL(V ) normalizes
E while permuting the members of both F( X (V )) and F(Y (V )).
For each v ∈ V , there is some αv = ±1 ∈ R and some v˜ ∈ V such that
ev g = αv ev˜ . Also for each a ∈ V , there is some βa = ±1 ∈ R and some a˜ ∈ V
such that e∗a g = βa e∗a˜ . By linearity,
X
X
a·v
a·v
αv (−1 ) ev˜ =
(−1 ) ev g
v
v
=
2(m+1)/2e∗a g
= 2(m+1)/2βa e∗a˜
X
(−1 )a˜·v ev ,
= βa
v
so that αv (−1 )a·v = βa (−1 )a˜·v˜ for all a, v ∈ V .
If v = 0, then v˜ = 0 and αv = 1, so that βa = 1 for all a ∈ V . Similarly αv = 1
for all v ∈ V , and hence a · v = a˜ · v˜ for all a, v ∈ V . If v, w ∈ V , then for all
a ∈ V,
a˜ · v g
− w = a · (v − w) = a˜ · (v˜ − w)
˜
so that vg
− w = v˜ − w˜ . It follows that the permutation v → v˜ is linear! Let A be
a matrix such that v˜ = v A for all v. Then a˜ · v A = a · v for all v ∈ V , and this
equation uniquely determines a˜ in terms of a and A . (In fact a˜ = a A −T .)
We digress with an observation concerning orthogonal spreads. While we
continue to assume that our field has order 2, the following result and its proof
hold for any finite field GF(q ).
L EMMA 3.15. Let 6 be a spread in an O+ (2m + 2, 2 )-space V , where m > 1.
If h ∈ O+ (2m + 2, 2 ) induces the identity on 6, then h = 1.
Proof. Otherwise, we may assume that |h| = p is prime. Consider three different
subspaces E , F, F 0 ∈ 6. Since the bilinear form on V induces an isomorphism
from F 0 to the dual space of E , the representations of h on E and F 0 are
contragredient. So are those on F and F 0 . Thus, the representations on E and F
are isomorphic.
Let W be the sum of an isomorphism class of h-isomorphic submodules of
E . Then W is isomorphic to a submodule W 0 of F . If U is any irreducible
submodule of V isomorphic to a submodule of W , then U ⊆ W + W 0 : each u ∈ U
can be written u = e + f with e ∈ E , f ∈ F , and u 7→ e defines an isomorphism
from U into E and hence into W . Similarly, f ∈ W 0 .
450
A . R . CALDERBANK
et al.
Thus, if dim W = k then the 2k -space W + W 0 meets each member of 6 in a k space. Since these intersections pairwise meet at 0, we have 22k − 1 > (2k − 1 )|6|,
so that 2k + 1 > 2m + 1 and hence k > m.
If p = 2 then we can choose W to be the space of fixed vectors within E .
Then CV (h ) has dimension 2 dim W = 2m, so that [ V, h] = CV (h )⊥ has dimension
2. If e 6= eh ∈ E then e − eh ∈ [ V, h]. Thus, the 2-space [ V, h] meets the 2m + 1
members of 6 in different points, which is impossible since m > 1.
Thus, h is semisimple. However, 2m > m + 1, so there cannot be two different
choices for W in E , and hence h is irreducible on E . Then p | (2 (m+1)/2 + 1 ).
Moreover, a Sylow p-subgroup of O+ (2m + 2, 2 ) is cyclic. An element of order
p can be obtained as follows. We may assume that V = GF(2m+1 ) ⊕ GF(2m+1 ),
equipped with the quadratic form ϕ(x, y ) := T (xy ), where T : GF (2m+1 ) → GF(2 )
is the trace map. We may also assume that h : (x, y ) 7→ (α x, α−1 y ) with
(m+1 )/2
, the following n-spaces are fixed
α ∈ GF(2m+1 ) of order p. Since α−1 = α2
(m+1 )/2
by h: GF (2m+1 ) × 0 and {(x, ax2
) | x ∈ GF (2m+1 )} for each a ∈ GF (2m+1 ).
Since they cover all of V , these are all of the (m + 1 )-spaces fixed by h. We
need to determine how many of them are totally singular. Let a ∈ GF(2m+1 ).
(m+1 )/2
Then T (axx2
) = 0 for all x ∈ GF(2m+1 ) if and only if T (a GF(2(m+1)/2 )) = 0,
and this occurs for only 2 (m+1)/2 choices of a. Thus, h fixes only 2 (m+1)/2 + 1
totally singular (m + 1 )-spaces. Since 2(m+1)/2 + 1 < |6|, we have the desired
contradiction.
P ROPOSITION 3.16. Let 61 and 62 be orthogonal spreads of E.
m +1
(i) Any isometry of E sending 61 to 62 is induced by an element of O (R2 )
sending F(61 ) to F(62 ) (in fact by many such elements). In particular, the
m +1
set-stabilizer of 61 in + (2m + 2, 2 ) is induced by a subgroup of O (R2 )
preserving the set F(61 ).
m +1
(ii) The subgroup E is the pointwise stabilizer of F(61 ) in O (R2
).
2m+1
(iii) Any element of O (R
) sending F(61 ) to F(62 ) normalizes E and
induces an orthogonal transformation of E sending 61 to 62 .
Proof. (i) This follows immediately from the definition of F(61 ) in Theorem 3.4.
(ii) By Lemma 3.14, the pointwise stabilizer of F(61 ) normalizes E and
hence induces a group of orthogonal transformations on E . By Lemma 3.15, the
pointwise stabilizer of 61 in O+ (2m + 2, 2 ) is 1.
(iii) The sets F( A ), with A ∈ 61 , are the maximal families of pairwise
perpendicular members of F(61 ) (this reflects the graph structure of F(61 )).
m +1
Part (ii) implies that, if g ∈ O (R2 ) sends F(61 ) to F(62 ), then g normalizes
E . Thus, g induces an automorphism on E .
In view of Corollary 10.9√below, it follows that, if m is odd and composite,
then there are more than 2 m/2 extremal line-sets F(6) that are inequivalent
m +1
under the action of O (R2 ).
Z4 - KERDOCK CODES
451
4. Extraspecial 2-groups continued
Let k = m in x 2, and let E m be the extraspecial 2-group of order 21+2m
introduced there. No parity assumption is made concerning m. We are now going
to extend E m slightly, at the same time switching from real to complex space.
In this section we will view ev , with v ∈ V , as the standard basis of C N , where
N = 2m . We equip C N with the usual hermitian inner product, and denote the full
isometry group by U (C N ). This inner product restricts to the usual one on the
N
N
N
real ‘subspace’
√ hev | v ∈ V iR, so that O (R ) < U (C ). Moreover, E m < U (C ).
Let i = −1 and F := E m hiI i (this is the central product of E m and hiI i: these
two groups commute, and they intersect in the centre h− I i of E m ; cf. Aschbacher
[1, p. 32]). Then | F | = 22+2m , Z ( F ) = hiI i, and F := F / Z ( F ) is elementary
abelian of order 22m . As before we will use an overbar to denote the natural
map F → F / Z ( F ). Once again we identify h− I i with Z2 in order to obtain a
non-singular alternating binary form on F defined by ( f 1 , f 2 ) F = [ f 1, f 2 ] ∈ h− I i
for f 1 , f 2 ∈ F. As in (2.5), ( X (a )Y (b ), X (a 0 )Y (b 0 )) F = a · b 0 − a 0 · b for all
a, b, a0, b0 ∈ V . This time the m-spaces X (V ) and Y (V ) are totally isotropic: each
is perpendicular to itself. Moreover, X (V ) ∩ Y (V ) = 0.
m
Once again we will require many isometries of C2 that normalize F , as well
as the isometries they induce on F by conjugation. Again let v1 , . . . , vm denote
the standard basis of V = Zm
2 , and let x j = X (v j ) and y j = Y (v j ). Then x1 , . . . , xm
and y1 , . . . , ym are dual bases of X and Y , and x1 , . . . , xm , y1 , . . . , ym is a basis of
F . The group L introduced in (2.13) lies in U (C N ) since it lies in O (R N ), and it
normalizes F since it normalizes Em and hiI i.
In order to obtain additional isometries we now turn to the counterparts of the
diagonal transformations d M defined in (2.10), thereby finally introducing Z4 into
these considerations. This time it is straightforward to check that the isometries
of F which induce the identity on Y (V ) are precisely those described by matrices
I P
, where P is a binary symmetric m × m matrix (compare Lemmas 2.12 and
O I
m
4.7). For a given
P we require an isometry d P of C2 that normalizes F and
induces OI PI on F , by analogy with (2.11).
In place of the quadratic form Q M : V → Z2 that occurs in (2.10), we will use
a map TP : V → Z4 which is called a Z4 -valued quadratic form on V by Brown
[7] (cf. [35]). This is obtained as follows. Since P = ( Pjk ) is a (0, 1 )-matrix, we
may view its entries 0, 1 as elements of Z4 . Start with b
V = Zm
4 and with the
T
quadratic form b
V ; we emphasize that the entries of P are to be
v Pb
v for b
v∈b
viewed as 0, 1 ∈ Z4 .
We say that b
V is a lift of v ∈ V if b
v = (α1 , . . . , αm ) ∈ b
v ≡ v (mod 2 ). Define
X
X
(4.1)
TP (v) :=
Pjj α2j + 2
Pjk α j αk ,
j
j <k
where b
v = (α1 , . . . , αm ) is some lift of v; note that (4.1) is independent of the
choice of lift. It is immediate that, for any u, v ∈ V ,
TP (u + v) = TP (u ) + TP (v) + 2b
uP b
vT ,
(4.2)
where b
u and b
v are lifts of u and v, respectively.
Recall that two binary quadratic forms are associated with the same alternating
bilinear form if and only if their difference is a linear functional. An analogous
result holds here.
452
A . R . CALDERBANK
et al.
L EMMA 4.3. A map TP0 : V → Z4 satisfies the condition in (4.2 ) if and only if
v Db
v T for a unique m × m diagonal (0, 1 )-matrix D.
= TP (v) + 2b
TP0 (v)
Proof. Since TP − TP0 satisfies
(TP − TP0 )(u + v) = (TP − TP0 )(u ) + (TP − TP0 )(v)
for all u, v ∈ V , it is an additive map V → Z4 and thus maps into 2Z4 . When
2Z4 is identified with Z2 we see that TP − TP0 is a linear functional on V . Any
such linear functional into 2Z4 can be written as v 7→ 2b
v Db
v T for a unique m × m
diagonal (0, 1 )-matrix D .
By analogy with (2.10), let
d P := diag[i TP (v) ].
(4.4)
Then
ev d −P 1 X (a )d P X (a ) = i−TP (v) ev+a d P X (a )
= i−TP (v) i TP (v+a ) ev
= i TP (a ) i2a Pv ev
= i TP (a ) (−1 )a P·v ev ,
T
so that d −P 1 X (a )d P X (a ) = i TP (a ) Y (a P ). Since d P commutes with Y (b ) (both are
diagonal matrices), we have
I P
−1
(4.5)
d P ( X (a )Y (b ))d P = X (a )Y (a P )Y (b ) = X (a )Y (b )
.
O I
We now have enough isometries of F to generate the symplectic group
m
Sp(2m, 2 ) using conjugation by elements of U (C2 ) that normalize F . Namely, if
L \ := h F, GL (V ), H , d P | P is a binary symmetric m × m matrixi ,
\
(4.6)
\
then L contains F as a normal subgroup, and L / F acts on F by conjugation,
inducing Sp(2m, 2 ) on F (cf. [33, p. 72]). The analogue of Lemma 2.12 is the
following lemma.
L EMMA 4.7. (i) Every totally isotropic m-space W of F such that Y (V ) ∩ W = 0
has the form
I P
W = d −P 1 X (V )d P = X (V )
= { X (a )Y (a P ) | a ∈ V }
O I
for a unique binary
symmetric m × m matrix P. The linear transformation of E
produced by OI PI preserves the form ( , ).
(ii) Let P1 and P2 be binary symmetric m × m matrices for which the
corresponding totally isotropic m-spaces W1 and W2 satisfy Y (V ) ∩ W1 = Y (V ) ∩
W2 = 0. Then W1 ∩ W2 = 0 if and only if P1 − P2 is non-singular.
5. Z4 -Kerdock codes, symplectic spreads and complex line-sets
with prescribed angles
We continue with the notation introduced in x 4.
Z4 - KERDOCK CODES
453
D EFINITION . A symplectic spread of F is a family 60 of 2m + 1 totally
isotropic m-spaces such that every point in F lies in a unique member of 60 .
(Note that there are exactly (2m + 1 )(2m − 1 ) points, 2m − 1 of which are in any
given m-space.)
A symplectic spread is a√spread in the conventional √
sense (see Dembowski
[12, p. 219]): a family 60 of |W | + 1 subspaces of size |W | of a finite vector
0
space W such that every non-zero vector lies in exactly
√ one member of 6 . A
0
0
spread 6 determines an affine plane A(6 ) of order |W | as follows: points are
vectors, and lines are the cosets A + w, where A ∈ 60 and w ∈ W . This accounts
for the importance of spreads in finite geometry.
After replacing 60 by 60 ` for some ` ∈ L \ we may assume X (V ), Y (V ) ∈ 60 .
By Lemma 4.7, any totally isotropic
m-space A ∈ 60 , with A 6= Y (V ), can be
I PA
written in the form X (V ) O I = { X (a )Y (a PA ) | a ∈ V } for a unique binary
symmetric m × m matrix PA ; here, OI PIA induces an isometry of F . In terms of
the affine plane A(60 ), the subspace { X (a )Y (a PA ) | a ∈ V } can be thought of as
the line ‘ y = x PA ’.
Part (ii) of Lemma 4.7 implies that
S(60 ) := { PA | A ∈ 60 \ {Y (V )}}
(5.1)
is a set of 2 symmetric m × m matrices such that the difference of any two is
non-singular. If the word ‘symmetric’ is deleted here, the preceding condition on
m × m matrices is essentially the definition of a spread set in [12, p. 220]; the
corresponding affine plane (S(60 )) can be viewed as having V ⊕ V as its set
of points, while the lines are the sets of points (x, y ) ∈ V ⊕ V having the familiar
appearance
m
A
x=b
or
y = x PA + b
with b ∈ V and A ∈ 60 \ {Y (V )}.
(5.2)
On the other hand, the corresponding Z4 -code K4 (60 ) comprises 22m+2 vectors in
m
Z24 ; the 2m coordinate positions are labelled by vectors v in V = Z2m . Much of
m
the rest of this paper is concerned with the following subset of Z24 :
s ·b
s∈b
V , ε ∈ Z4 },
v + ε)v | A ∈ 60 , b
K4 (60 ) := {(TPA (v) + 2b
where TPA is the Z4 -valued quadratic form introduced in (4.1) with P = PA , and
v ∈ Z4m is congruent to v ∈ V modulo 2 (again see x 4). Alternatively, we can
b
write
(5.3)
s ·b
v + ε)v | A ∈ 60 , s ∈ V, ε ∈ Z4 }.
K4 (60 ) := {(TPA (v) + 2b
If m is odd, K4 (60 ) will be called a Z4 -Kerdock code.
R EMARK . It is again important to observe that this notation is ambiguous. The
set { PA | A ∈ 60 \ {Y (V )}} depends on the choice of a pair of members of 60
to play the role of X (V ), Y (V ), and on the choice of dual bases in X (V ) and
Y (V ). Thus, the notation S(60 ) for the preceding set of matrices is ambiguous.
Up to equivalence of codes, K4 (60 ) depends only on 60 and on the distinguished
member Y (V ) of 60 that was discarded in the construction of the set S(60 ).
Next we consider the unitary geometry of the 2-group F . The next two
lemmas are the counterparts of Lemmas 3.2 and 3.3.
454
A . R . CALDERBANK
et al.
L EMMA 5.4. (i) The set {hev i | v ∈ V } is the set of irreducible submodules for
Y (V ).
(ii) The set {heb∗ i | b ∈ V } is the set of irreducible submodules for X (V ).
In fact, this is immediate by Lemma 3.2, since Y (V ) and X (V ) are real matrix
groups. However, when these are moved to other subgroups of F using L \ ,
complex numbers enter.
L EMMA 5.5. Let A and B be subgroups of F such that A and B are totally
isotropic m-dimensional subspaces of the symplectic space F.
m
(i) The set F( A ) of A-irreducible subspaces of C2 is an orthogonal frame.
(ii) Assume that A ∩ B = 0, and let u 1 and u 2 be unit vectors in different
members of F( A ) ∪ F( B ). If u 1 and u 2 are both in F( A ), or are both in F( B ),
then (u 1 , u 2 ) = 0; otherwise, |(u 1 , u 2 )| = 2−m/2 .
(iii) The set F( A ) is left invariant by F.
(iv) If B = Y (V ), and if u 1 and u 2 are unit vectors such that u 1 lies in a member
of F( A ) and u 2 lies in a member of F( B ), then (u 1 , u 2 ) ∈ {1, −1, i, −i}2−m/2 .
Proof. For parts (i)–(iii), by using the group L \ we may assume that A = X (V )
and B = Y (V ). Parts (i) and (ii) follow directly from Lemma 5.4 as in the proof
of Lemma 3.3. Part (iii) follows from the fact that h A , iI i is normal in F .
(iv) By Lemma 5.4(ii) and (4.4), each member of F( A ) = F( X (V ))d P is
spanned by a unit vector all of whose coordinates are 1, −1, i, or −i.
T HEOREM 5.6. Let 60 be any symplectic spread of the symplectic space F.
Then
[
F(60 ) :=
F( A )
A ∈60
m
consists of 2m (2m + 1 ) 1-spaces in C2 such that, if u 1 and u 2 are unit vectors in
different members of F(60 ), then |(u1 , u2 )| = 0 or 2−m/2.
This follows immediately from the preceding lemma. As in Corollary 3.5, the
Z4 -code K4 (60 ) introduced in (5.3) can be recovered very simply from F(60 ).
C OROLLARY 5.7. K4 (60 ) = {(cv )v ∈ Z24 | h(icv )v i ∈ F(60 )} .
m
Proof. Use (2.8), (4.4) and (5.3) as in the proof of Corollary 3.5.
Bounds for lines-sets in C N with prescribed angles
Once again the line-set F(60 ) is extremal in the sense that |F(60 )| meets
another upper bound obtained by Delsarte, Goethals and Seidel [11], this time for
line-sets in C N with prescribed angles. The proofs are very similar to those in x 3.
For any N let e1 , . . . , e N be the standard basis of C N . Once again equip C N
with the usual hermitian inner product, and let S denote the unit sphere. Let
a∗ denote the complex conjugate of a ∈ C N . Consider a spanning set  of n
points of S such that |(a, b )| ∈ {0, α} for all a 6= b in , where 0 < α < 1. Then
Gram() = I + αC, where C is a matrix with diagonal entries 0 and all other
Z4 - KERDOCK CODES
455
non-zero entries u satisfying |u| = 1. Once again the spectrum of C has the form
{(−1/α)n− N , λ1 , . . . , λ N }. Since Gram () is a hermitian matrix, its eigenvalues
λ1 , . . . , λ N are real.
The vector space S2 (C N ) ⊗ C N has dimension N N 2+1 . For each a ∈  this
P
space contains the tensors va := a ⊗ a ⊗ a∗ and ha := 12 i (a ⊗ ei + ei ⊗ a ) ⊗ ei .
As before,
!
I + α3 C
I + αC
Gram (va ; ha | a ∈ ) =
,
I + αC 12 ( N + 1 )( I + αC )
and I + α3 C is positive definite, from which we deduce the absolute bound
N+1
.
n<N
2
Using row and column transformations as in x 3, we find that
0 < 1 + αλi <
( N + 1 )(1 − α2 )
2 − ( N + 1 )α2
(5.8)
assuming that the denominator is positive, and hence
n = Tr ( I + αC ) =
N
X
i =1
(1 + αλi ) <
N ( N + 1 )(1 − α2 )
,
2 − ( N + 1 )α2
(5.9)
which is the special bound obtained by Delsarte, Goethals and Seidel.
When equality holds in the special bound, the matrix C has two eigenvalues:
−1/α with multiplicity n − N and β := (n − N )/ N α. In this case we consider
the Gram matrix Gram(a ⊗ a∗ ) of the tensors a ⊗ a∗ ∈ C N ⊗ C N for a ∈ .
This time Gram (a ⊗ a∗ ) = I + α2 A , where A is a (0, 1 )-matrix whose row
sums are all β/α (the diagonal entries of CC ∗ ). As in x 3 we find that the
eigenvalues of A are β/α with multiplicity 1, −1/α2 with multiplicity n − N 2 ,
and ( N α2 − 1 )/(2α2 − ( N + 1 )α4 ) with multiplicity N 2 − 1. It follows that A is
the adjacency matrix of a strongly regular graph. In the special ‘Kerdock case’
α2 = 1/ N , the graph is the union of N + 1 disjoint N -cliques:  is the union of
N + 1 orthonormal bases, with members of different bases not perpendicular.
R EMARKS . (1) In the special ‘Kerdock case’, the graph-theoretic structure of
any extremal line-set meeting the special bound coincides with that of any set
F(60 ) derived from a symplectic spread 60 , just as we saw in Proposition 3.5
for the real case. However, the difficult open questions posed in the remarks
following that proposition are even more difficult for the present type of extremal
line-set. Namely, once again an extremal line-set produces a Gram matrix whose
off-diagonal blocks are Hadamard matrices, but this time these are complex
Hadamard matrices. Their entries may involve complex numbers other than
{1, −1, i, −i}, since it is permitted to multiply any row by a complex number of
modulus 1 and the corresponding column by its complex conjugate. However, it
is not clear whether we can make all entries lie in {1, −1, i, −i} by appropriate
row and column multiplications of this form, as holds for F(6). Moreover, we
do not even have a characterization of the sets F (6) in terms of their Z4 -codes,
as we did in the binary case (see the remarks following Proposition 3.12).
456
A . R . CALDERBANK
et al.
(2) Delsarte, Goethals, and Seidel obtained their bounds on the cardinality of
systems of lines with prescribed angles using Jacobi polynomials: absolute bounds
that depend only on the number of possible angles, and special bounds that depend
on the angles themselves. Theorem 7.5 of their paper asserts that, if equality holds
in the special bound, then the extremal line-sets form an association scheme. In
the special case of two possible angles α, β, the extremal line-sets form a strongly
regular graph, where two lines are adjacent if the angle between them is β. Our
derivation of a special case of this result uses only elementary tensor algebra,
and in the special ‘Kerdock case’ it reveals the simple structure of the strongly
regular graph. For connections to homogeneous and harmonic polynomials on the
sphere we refer the reader to Koornwinder [23, 24].
(3) Is it possible to say something about the Molien series of the groups L and
L \ , such as the minimal degree of an invariant? We would then have information
regarding properties of F(6) and F(60 ) as spherical designs (see Seidel [30] for
background).
The geometry of the line-sets F(60 )
The counterparts of Lemma 3.14 and Proposition 3.16 are as follows.
m
L EMMA 5.10. The group of all elements of U (C2 ) sending each of the frames
F( X (V )) and F(Y (V )) to itself normalizes F.
P ROPOSITION 5.11. Let 610 and 620 be symplectic spreads of F.
(i) Any symplectic transformation sending 610 to 620 is induced by an element of
m
U (C2 ) sending F(610 ) to F(620 ) (in fact by many such elements). In particular,
m
the set-stabilizer of 610 in Sp (2m, 2 ) is induced by a subgroup of U (C2 )
preserving F(610 ).
m
(ii) The pointwise stabilizer of F(610 ) in U (C2 ) is h F, C I i, where C ⊆ C∗ is
the unit circle; moreover, F is the set of all elements in this pointwise stabilizer
of order dividing 4.
m
(iii) Any element of U (C2 ) sending F(610 ) to F(620 ) normalizes F and induces
a symplectic transformation of F sending 610 to 620 .
Proof. (i) This follows from Theorem 5.6.
m
(ii) By Proposition 5.10, the pointwise stabilizer of F(610 ) in U (C2 ) normalizes
F and hence induces a group of symplectic transformations on F . As in the proof
of Proposition 3.16, the pointwise stabilizer of 610 in Sp (2m, 2 ) is 1.
The unit circle enters as the scalars lying in the unitary group. The last part of
(ii) follows from the fact that the product of an element of F with a scalar of
order not dividing 4 also has order not dividing 4.
(iii) The sets F( A ), with A ∈ 61 , are the maximal families of pairwise
m
perpendicular members of F(610 ). Part (ii) implies that, if g ∈ U (C2 ) sends
F(601 ) to F(602 ), then g normalizes h F, C I i. Thus, g induces a symplectic
transformation of h F, C I i/ Z (h F, C I i ) ∼
= F.
In view of Corollary 10.9√ below, it follows that, if m is odd and composite,
then there are more than 2 m/2 extremal line-sets F(60 ) that are inequivalent
m
under the action of U (C2 ).
457
Z4 - KERDOCK CODES
m +1
6. From the real space R2
m
to the complex space C2
+1
Let m be an odd integer and let V = Zm
and E = Em+1 be as in x 3. Recall
2
that v1 , . . . , vm+1 is the standard basis of V .
Fix an element ω ∈ E of order 4, so that ω is a non-singular vector in
E . Writing ω = X (a )Y (b )(− I )γ as in Lemma 2.1, we see from (2.3) that its
centralizer is
0
C E (ω) = { X (a0 )Y (b0 )(− I )γ | a0 , b0 ∈ V, γ 0 ∈ Z2 , and a · b0 = a0 · b},
so that C E (ω) projects onto ω ⊥ by (2.5).
m +1
Since ω2 = − I , the eigenvalues of ω are ±i, so ω fixes no 1-space of R2 .
In fact, for every vector u we have (u, uω) = 0 (since (u, uω) = (uω, uω2 ) =
−(u, uω)). Choose ω so that ω = X (a )Y (b ) (cf. Lemma 2.1). Then a · b = 1 by
(2.4). Let V 0 be any hyperplane of V not containing a. By the definition of X (a )
and Y (b ) (cf. x 2), for any v0 ∈ V 0 we have ev0 ω = ±ev0 +a , where v0 + a ∈
/ V 0. It
m +1
0
0
2
follows that {ev0 , ev0 ω | v ∈ V } is an orthonormal basis of R .
Also since ω2 = − I , we may identify the ring R + ωR with C. In view of
the conjugacy of elements of order 4 in E as expressed in Lemma 2.9(ii), for
simplicity we may assume that ω = X (vm+1 )Y (vm+1 ). However, it is important to
note that there are many different choices for fields isomorphic to C obtained in
this manner. For, we will have a specific orthogonal spread 6 of E , so that we
cannot freely change to a different copy of C while preserving 6. (Note. Overbars
will continue to have the same meaning as before; they will never be used to
denote complex conjugation.)
Again for simplicity we choose V 0 := hv1 , . . . , vm i. If we let X (V 0 ) = { X (b ) |
b ∈ V 0} and define Y (V 0 ) similarly, then Em := X (V 0 )Y (V 0 )h− I i is an extraspecial
group of order 21+2m , and F := C E (ω) is just Em hωi, as in x 5.
m +1
m
Now we can view R2 as C2 . Then we have seen that {ev0 | v0 ∈ V 0 } is a basis
over C. Use this as an orthonormal basis over C in order to define a hermitian
inner product. In an attempt to decrease the likelihood of confusion, throughout
the remainder of this section we will use the notation ( , )R and ( , )C for the
real and hermitian inner products on our vector space. Then (u, u )R = (u, u )C for
all vectors u, perpendicularity with respect to ( , )C implies perpendicularity with
respect to ( , )R (but not vice versa), and F preserves ( , )C .
If A is a totally singular (m + 1 )-space of the orthogonal space E , then the
frame F( A ) produces a set
Fω ( A ) := {Cu | Ru ∈ F( A )}
m
of 1-spaces of C2 . Lemma 3.3(iii) implies that ω permutes the members of
F( A ). Since ω cannot fix any member of F( A ), it must map each member to
another one perpendicular to the first; together these two span a 1-space over C.
Any two complex 1-spaces obtained in this manner are perpendicular: Fω ( A ) is
m
a frame of C2 .
P ROPOSITION 6.1. Let 6 be an orthogonal spread of E, and let F(6) be the
m +1
corresponding set of lines in R2 . Then
[
Fω (6) :=
Fω ( A )
A∈6
458
A . R . CALDERBANK
et al.
m
is a set of 2m (2m + 1 ) lines in C2 . If u and v are unit vectors in different
members of Fω (6), then |(u, v)C | = 0 or 2−m/2 .
Proof. We may assume that u and v are in members of Fω ( A ) and Fω ( B )
respectively, where A and B are distinct members of 6. Let ui , with 1 < i < 2m ,
be unit vectors in different members of Fω ( A ). Then u1 , . . . , u 2m , u 1 ω, . . . , u 2m ω
is a real orthonormal basis (since (Cui , Cu j )C = 0 for i 6= j, and we know that
(ui , ui ω)R = 0). Write
v=
2m
X
j =1
u j (a j + b j ω) =
2m
X
(a j u j + b j (u j ω))
j =1
for some a j , b j ∈ R. By Theorem 3.4, a j = (v, u j )R and b j = (v, u j ω)R are
±2−(m+1)/2. It follows that |(v, u j )C | = (a2j + b2j )1/2 = 2−m/2 . Since u = αu j for
some j and some scalar α of norm 1, we have |(v, u )C | = 2−m/2.
7. From orthogonal spreads to symplectic spreads and back
We have yet to see how the line-set Fω (6) introduced in x 6 relates to
symplectic spreads. Recall that ω = X (vm+1 )Y (vm+1 ) has order 4.
⊥
Note that, if A ∈ 6, then A 6⊆ ω⊥ ; for otherwise ω ∈ A = A, which contradicts
⊥
the non-singularity of the ω. Hence dim( A ∩ ω ) = m. Every singular vector in
ω⊥ is in a unique member of { A ∩ ω⊥ | A ∈ 6}. Now project into the symplectic
2m-space F = ω ⊥ /hωi in order to obtain the set
6ω := {h A ∩ ω ⊥ , ωi/hω i | A ∈ 6}.
(7.1)
This is a family of 2m + 1 totally isotropic m-spaces covering all of F : it is a
symplectic spread.
Kantor [16] described a simple construction reversing this process. Start with
a symplectic spread 60 of F = ω ⊥ /hωi. This lifts to a set of totally singular
m-spaces of ω ⊥ , namely
600 = {W | W is a totally singular subspace of ω⊥
that projects onto a member of 60 }.
We may assume that one member of 600 is contained in Y (V ). Every other
member of 600 is contained in exactly two totally singular (m + 1 )-spaces of E ;
exactly one of those meets Y (V ) only in 0, and we choose this one. Let 6 be the
set of all of the totally singular (m + 1 )-spaces of E arising in this way (including
Y (V )):
6 := {Y (V )} ∪ { X | X is a totally singular subspace of E
that contains some member of 600
(7.2)
and has only 0 in common with Y (V )}.
Then 6 is an orthogonal spread of E . Moreover, 6ω = 60 in the notation of the
previous paragraph. We emphasize that this spread 6ω depends on the choice of
ω. Given 6, different choices of ω can produce inequivalent symplectic spreads
Z4 - KERDOCK CODES
459
and inequivalent Kerdock codes. This is studied at length in [17]. Even when 60
produces a desarguesian plane, after constructing 6 one can then make different
choices of non-singular points ω in order to obtain new planes and codes (this is
crucial for Theorem 10.3 below).
We will also need a different and more computational way of viewing this
process, in terms of the matrices appearing in Lemmas 2.12 and 4.7.
Consider a totally isotropic m-space W 0 of the symplectic space F such
that W 0 ∩ Y (V 0 ) = 0. Let P be the binary symmetric m × m matrix such that
W 0 = X (V 0 ) OI PI (cf. Lemma 4.7).
L EMMA 7.3. There is a unique totally singular (m + 1 )-space W = X (V ) OI MI
of E such that W ∩ Y (V ) = 0 and hW ∩ ω⊥ , ω i/hω i = W 0 . Here M is the binary
skew-symmetric (m + 1 ) × (m + 1 ) matrix given by
!
P + d ( P )T d ( P ) d ( P )T
(7.4)
M=
,
0
d ( P)
where d ( P ) is the vector in V 0 whose coordinates are the diagonal entries of P
in the natural order.
R EMARK . Equation (7.4) defines a bijection P 7→ M from symmetric m × m
matrices P to skew-symmetric (m + 1 ) × (m + 1 ) matrices M . Indeed, given a
skew-symmetric (m + 1 ) × (m + 1 ) matrix M , let the last row be (d 0 ) and find
P from the principal minor indicated in (7.4). Conversely, given a symmetric
m × m matrix P, observe that the matrix M in (7.4) is, indeed, skew-symmetric.
However, note that this bijection is not linear.
+1
in the form (v, α) with
Proof of Lemma 7.3. Write vectors of V = Zm
2
and
.
Then
the
vectors
of
can
be
written (v, α, z, β) with
E
v ∈ V 0 = Zm
α
∈
Z
2
2
0
0
v, z ∈ V and α, β ∈ Z2 . Note that Y (V ) consists of all vectors having v = 0 and
α = 0. By (2.4), the quadratic form Q is given by Q (v, α, z, β) = v · z + αβ.
We are given the subspace W 0 = {(v, v P ) | v ∈ V 0} of F = X (V 0 )Y (V 0 ). Since
ω = X (vm+1 )Y (vm+1 ), we have ω = (0, 1, 0, 1 ) and hence ω⊥ = {(v, α, y, α) |
v, y ∈ V 0, α ∈ Z2 }. Pulling W 0 back to ω ⊥ yields a unique totally singular m-space
{(v, v · d ( P ), v P, v · d ( P )) | v ∈ V 0 }.
(For, this is an m-space, and is totally singular since v · v P = v · d ( P ) = (v · d ( P ))2
for all v ∈ V 0 .) Any totally singular (m + 1 )-space W properly containing this
m-space must contain a vector of the form (0, α, y, β); this vector is singular if
and only if α or β is 0. Since we want to have W ∩ Y (V ) = 0, we must have
α = 1 and β = 0.
Since (0, 1, y, 0 ) and (v, v · d ( P ), v P, v · d ( P )) need to be perpendicular for all
v ∈ V 0, it follows that y · v = v · d ( P ) and hence y = d ( P ). Thus, W is uniquely
determined (as we already observed above):
W = {(v, v · d ( P ) + α, v P + αd ( P ), v · d ( P )) | v ∈ V 0, α ∈ Z2 }
= {(v, γ, v P + vd ( P )T d ( P ) + γ d ( P ), v · d ( P )) | v ∈ V 0, γ ∈ Z2 }
since [v · d ( P )]d ( P ) = vd ( P )T d ( P ).
460
A . R . CALDERBANK
et al.
Note that the parity of m did not enter into the lemma. Also, there is a simple
counterpart of this lemma for any perfect field of characteristic 2: just let the
entries of d ( P ) be the square roots of the diagonal entries of P in the natural
order.
Now consider a symplectic spread 60 of F containing
X (V 0 ) and Y (V 0 ).
I PA
0
Let 6 \ {Y (V )} consist of the subspaces A = X (V ) O I for symmetric m × m
matrices PA . Then Lemma 7.3 lifts 60 to the following orthogonal spread:
I M PA 0
0
(7.5)
A
Y
V
6 = {Y (V )} ∪ X (V )
6
(
)
∈
\
{
}
O
I
where
PA + d ( PA )T d ( PA ) d ( PA )T
MA =
d ( PA )
0
!
.
(7.6)
Finally, we return to the complex line-set Fw (6) defined in Proposition 6.1.
P ROPOSITION 7.7. Let 6 be an orthogonal spread in E, and let ω ∈ E have
order 4. Then Fω (6) = F(6ω ).
Proof. Let A ∈ 6. Let A ω denote the preimage in F of the member A ∩ ω⊥
of the symplectic spread 6ω . It suffices to show that Fω ( A ) = F( A ω ), where the
latter frame is calculated using 6ω .
Note that A ω = C A (ω)hω I i. Since A fixes each member of F( A ), so does
C A (ω). Also, ω merges pairs of members of F( A ) into 1-spaces over C, and
these pairs must be C A (ω)-invariant. Hence A ω fixes each 1-space in Fω ( A ). By
m
Lemma 5.5(ii), there are exactly 2m 1-spaces of C2 fixed by A ω ; we have found
them all.
R EMARKS . (1) By (7.2) and Lemmas 2.12(ii) and 4.7(ii), the correspondence
(7.4) between binary symmetric m × m matrices P and binary skew-symmetric
(m + 1 ) × (m + 1 ) matrices M commutes with the action of the general linear
group GL(V 0 ). This is easy to verify directly, as shown below, and is to be
expected since the matrix P depends on the choice of dual basis for X (V 0 ),
Y (V 0 ), and GL(V 0 ) is transitive on such dual bases:
!
P + d ( P )T d ( P ) d ( P )T
(7.4)
M=
P
0
d ( P)
-
P 7→ A PA
M 7→
T
?
A PA T
(7.4)
-
?
A 0
0 1
A PA + Ad ( P ) d ( P ) A
Ad ( P )T
d ( P) AT
0
T
T
M
!
AT
0
0
1
461
Z4 - KERDOCK CODES
Hence (7.4) determines a many-to-one correspondence between congruence classes
of symmetric m × m matrices under GL(V 0 ) and congruence classes of skewsymmetric (m + 1 ) × (m + 1 ) matrices under GL(V ).
Any skew-symmetric matrix is congruent to a block diagonal matrix Mk with
k diagonal blocks 01 10 and all other entries 0. Thus, two skew-symmetric
(m + 1 ) × (m + 1 ) matrices are congruent if and only if they have the same rank.
Any symmetric matrix is congruent to a block diagonal matrix P`, j with j
diagonal blocks 01 10 , ` − 2 j diagonal blocks (1) and all other entries 0. The
correspondence (7.4) maps the congruence classes containing P2k , j or P2k −1, j for
some j to the congruence class containing Mk .
(2) Let M1 and M2 be the binary skew-symmetric (m + 1 ) × (m + 1 ) matrices
obtained from the symmetric m × m matrices P1 , P2 respectively under the
correspondence (7.4). We claim that Rank( P1 − P2 ) = 1 or 2 if and only if
Rank( M1 − M2 ) = 2. This provides a realization of the ‘alternating forms graph’
[5, pp. 281–284, 287] in terms of symmetric matrices.
First suppose that Rank( P1 − P2 ) = 1 or 2. Then there exist v, z ∈ V 0 such that
P2 = P1 + ε 1v T v + ε 2 z T z + ε 3 (v T z + z T v)
for some ε 1 , ε 2 , ε 3 ∈ Z2 . Note that ε 3 = ε 2 ε1 if and only if Rank( P2 − P1 ) = 1 and
P2 = P1 + (ε1 v + ε 2 z )T (ε1 v + ε2 z ). For brevity, let d1 = d ( P1 ) ∈ V 0 be the vector
whose entries are the diagonal elements of P1 in the natural order. Then
!
P1 + d1T d1 d1T
M1 =
,
0
d1
!
P2 + (d1 + ε 1 v + ε 2 z )T (d1 + ε 1 v + ε 2 z ) (d1 + ε1 v + ε2 z )T
M2 =
,
0
d1 + ε1 v + ε2 z
and


(ε3 + ε 2 ε 1 )(v T z + z T v) + d1T (ε1 v + ε 2 z ) (ε1 v + ε 2 z )T
+(ε1 v T + ε 2 z T )d1
.
M2 + M1 = 
0
ε1v + ε2 z
If ε 3 = ε 2 ε 1 then
M2 + M1 =
d1T
1
(ε1 v + ε 2 z, 0 ) +
(ε1 v + ε 2 z )T
0
(d1 , 1 ),
and if ε 3 6= ε 2 ε 1 then M2 + M1 is similar to the matrix
T T v
z
(z, ε 1 ) .
(v, ε 2 ) +
ε2
ε1
In either case Rank( M2 − M1 ) = 2. It is not hard to show that this argument
reverses.
8. From Z4 -Kerdock codes to Z2 -Kerdock codes
In this section we show that the geometric map in x 7 from symplectic to
orthogonal spreads induces the Gray map from the corresponding Z4 -Kerdock
code to its binary image.
462
A . R . CALDERBANK
et al.
The Gray map
First we need to recall the definition of the Gray map used by Hammons et al.
in [14]. Figure 1 shows the Gray encoding of quaternary symbols 0, 1, 2, 3 in Z4
as pairs of binary digits.
F IG . 1. The Gray map from Z4 to Z22
The 2-adic expansion of c ∈ Z4 is
c = α(c ) + 2β(c );
(8.1)
write γ(c ) = α(c ) + β(c ) for all c ∈ Z4 . We now have three maps Z4 → Z2 , as
follows:
c α(c ) β(c ) γ(c )
0
1
2
3
0
1
0
1
0
0
1
1
0
1
1
0
They extend in an obvious way to maps from Z4N to Z2N for any positive
integer N . We construct binary codes from quaternary codes using the Gray map
φ : Z4N → Z22 N given by
φ(v) = (β(v), γ(v)),
where v ∈ Z4N .
(8.2)
Figure 1 is the case N = 1 of this.
The Lee weights of 0, 1, 2, 3 ∈ Z4 are 0, 1, 2, 1 respectively, and the Lee weight
wt L (a ) of a ∈ Z4N is the rational sum of the Lee weights of its components. This
weight function defines a distance d L ( , ) on Z4N called the Lee metric. The
fundamental property of the Gray map [14] is that it is an isometry (Z4N , Lee
metric) → (Z22 N , Hamming metric).
R EMARKS . (1) The Gray map φ distinguishes the partition π = {{i, i + N } |
i = 1, . . . , N } of the N coordinate positions. If we follow φ by an arbitrary
permutation h (which is, of course, an isometry of (Z22 N , Hamming metric))
then we obtain an isometry (Z4N , Lee metric) → (Z22 N , Hamming metric) that
distinguishes the partition πh . All isometries between these two metric spaces that
send 0 to 0 arise in this manner.
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Z4 - KERDOCK CODES
(2) Equivalences of codes over Z4 use the group of monomial transformations,
which preserves the Lee metric.
The Gray map and extraspecial 2-groups
We are now ready to relate spreads and extraspecial 2-groups to the Gray map.
First we need to collect the notation scattered throughout earlier sections.
Let m > 1 be an odd integer, and let E = E m+1 be the extraspecial 2group of order 21+2(m+1) studied in xx 2, 3 and 7. Let V 0 be the hyperplane
+1
spanned by the first m standard basis vectors. Fix ω =
hv1 , . . . , vm i of V = Zm
2
X (vm+1 )Y (vm+1 ) ∈ E of order 4, as in xx 6 and 7. Then E m := X (V 0 )Y (V 0 )h− I i
is an extraspecial group of order 21+2m , and we consider the slightly larger group
F = E m hω I i. Overbars will denote the natural map F → F / Z ( F ), not the natural
map E → E / Z ( E ). (Some care is needed with notation: E / Z ( E ) and F / Z ( F )
have dimension 2m + 2 and 2m, respectively.)
Start with a symplectic spread 60 of F . By (5.3) this determines the Z4 -code
0
s0 · vb0 + ε 0 )v0 ∈V 0 | A ∈ 60 , s0 ∈ V 0, ε 0 ∈ Z4 },
K4 (60 ) = {(TPA0 (v0 ) + 2b
where PA0 is the binary symmetric m × m matrix corresponding to the totally
0
isotropic m-space A ∈ 60 \{Y (V 0 )} via Lemma 4.7(i). (Note that the vector space
0
0
called ‘ V ’= Zm
2 in x 5 is now called ‘ V ’.) By (7.2), the symplectic spread 6 lifts
0
to an orthogonal spread 6 of E / Z ( E ); let b
A denote the lift of A ∈ 60 \{Y (V 0 )}.
(See the remark at the end of the last paragraph concerning the need for care with
notation: we cannot write ‘ A’ here.) In view of (3.1), 6 determines the Z2 -code
K(6) = {( Q M bA (v) + s · v + ε)v∈V | A Z / Z ∈ 6, s ∈ V, ε ∈ Z2 },
where M bA is the binary skew-symmetric (m + 1 ) × (m + 1 ) matrix corresponding
via Lemma 2.12(i) to the totally singular (m + 1 )-space b
A ∈ 6.
We view V = V 0 ⊕ Z2 as listed so that the first 2m coordinate positions are
(v0 0 ) and the rest are (v0 1 ). This order of coordinates determines our Gray map.
T HEOREM 8.3. The Gray map sends K4 (60 ) to K(6).
Proof. Consider the element (TPA0 (v0 ) + 2s0 · v0 + ε 0 )v0 ∈V 0 of K4 (60 ), where
A ∈ 60 , s0 ∈ V 0, ε 0 ∈ Z4 . In order to simplify notation write P = PA0 and
M = M bA . Recall from (4.1) that
0
TP (v0 ) + 2b
s 0 ·b
v 0 + ε0 =
m
X
P jj α2j + 2
X
P jk α j αk + 2b
s 0 ·b
v 0 + ε0 ,
(8.4)
j <k
j =1
0
where b
v 0 = (α1 , . . . , αm ) is any lift of v0 = (v10 , . . . , vm
) into Zm
4 . We need the
2-adic expansion of (8.4), so first we need to determine a, b ∈ Z2 such that
X
(8.5)
P jj α2j = a + 2b.
j
0
Passing modulo 2 gives d ( P ) · v = a since α2j ≡ α j (mod 2). (Of course, here
d ( P ) · v0 is calculated in Z2 .) Squaring (8.5) gives
X
X
P2jj α4j + 2
P jj Pkk α2j α2k = a2 = a ,
j
j <k
464
A . R . CALDERBANK
so that
2b =
X
P jj α2j −
j
and hence
b=
X
P2jj α4j − 2
j
X
P jj Pkk v0j v0k
et al.
X
P jj Pkk α2j αk2
j <k
(calculated in Z2 ),
j <k
since α4j = α2j for any α j ∈ Z4 . This produces the required 2-adic expansion
v 0 + ε0
TP (v0 ) + 2b
s 0 ·b
= (d ( P ) · v0 ⊕ λ) + 2
X
(8.6)
( P jk + P jj Pkk )α j αk + b
v 0 + λd ( P ) · b
v0 +µ ;
s 0 ·b
j <k
here ⊕ denotes binary addition (so that x + y = x ⊕ y + 2xy for x, y ∈ {0, 1} ⊂ Z4 ),
ε0 = λ + 2µ is the 2-adic expansion of ε 0 , and d ( P ) is identified with its lift. Let
U P be the ‘upper triangular portion’ of P + d ( P )T d ( P ), obtained by changing
to 0 all entries of the latter matrix below the main diagonal; this is a 0,1
matrix, with entries viewed over Z2 or Z4 , depending upon the context. Then
P + d ( P )T d ( P ) = U P + U PT , and we can rewrite (8.6) as
v 0 + ε0
TP (v0 ) + 2b
s 0 ·b
v 0 ⊕ λ) + 2(v
v 0T + b
v 0 + λd ( P ) · b
v 0) + µ .
s 0 ·b
= (d ( P ) · b
b0U Pb
(8.7)
By Lemma 7.3, the binary skew-symmetric (m + 1 ) × (m + 1 ) matrix M
corresponding to P is given by
!
!
!T
P + d ( P )T d ( P ) d ( P )T
U P d ( P )T
U P d ( P )T
.
M=
=
+
0
0
0
0
0
d ( P)
We are viewing V as the set of pairs (v0 ε) with v0 ∈ V 0 and ε ∈ Z2 . Then one of
the quadratic forms on V associated with M is given by
!
U P d ( P )T
0
(v0 ε)T + (s0 λ) · (v0 ε) + (λd ( P ) 0 ) · (v0 ε)
Q M (v) = (v ε)
0
0
= (v
b0U P v0 T + s0 · v0 + λd ( P ) · v0 ) + ε(v0 d ( P )T + λ).
0
0
(8.8)
It remains to compare (8.7) and (8.8). Each v ∈ V determines one coordinate
0
position for Zm
4 , with coordinate given by the left-hand side of (8.7). But v also
+1
determines two coordinate positions for V = Zm
, namely positions (v0 0 ) and
2
0
(v 1 ). By (8.7) and (8.8),
0
v 0 + ε 0 ∈ Z4 ;
in position v0 ∈ Zm
s 0 ·b
4 there is a coordinate TP (v ) + 2b
+1
this determines the following coordinates of a vector in Zm
:
2
in position (v0 0 ): v0U P v0 T + s0 · v0 + λd ( P ) · v0 + µ;
in position (v0 1 ): v0U P v0 T + d ( P ) · v0 + s0 · v0 + λd ( P ) · v0 + µ + λ.
In view of the 2-adic expansion (8.7),
(d ( P ) · v0 ⊕ λ) + 2(v
v 0T + b
v 0 + λd ( P ) · b
v 0 + µ)
s 0 ·b
b0 U Pb
produces
v0U P v0 T + s 0 · v0 + λ d ( P ) · v0 + µ
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Z4 - KERDOCK CODES
and
v0U P v0 T + s0 · v0 + λd ( P ) · v0 + d ( P ) · v0 + µ + λ,
which is exactly the behaviour of the Gray map (8.2).
In view of Corollaries 3.5 and 5.7, this theorem can also be regarded as
providing a transition from real to complex line-sets, as in Table 1.
This suggests another way to (try to) view the preceding theorem. Suppose that
the relationship between real and complex line-sets via extraspecial groups had
been noticed a few years ago. This would have suggested the definition of K4 (60 ),
by analogy with K(6). The desire to see a relationship between K4 (60 ) and
K(6) would then have led directly to the Gray map. This would not, however,
have made evident the many remarkable properties of this map discovered by
Hammons et al. [14].
In [14], a ‘Preparata’ code is constructed as follows: start with the classical
Kerdock code but in its Z4 -linear incarnation, form the dual code over Z4 , and
then pass back into the binary world. Therefore, it is important to understand the
meaning of Z4 -linearity of a Z4 -Kerdock code. This is dealt with in the following
proposition.
P ROPOSITION 8.9. The code K4 (60 ) is Z4 -linear if and only if S(60 ) = { PA0 |
A ∈ 60 } is closed under binary addition.
0
Proof. Let Q0 be the additive group of all of the maps T + ε : V 0 → Z4 such
that T is a Z4 -valued quadratic form and ε ∈ Z4 is a constant. This means that
T (u0 + v0 ) = T (u0 ) + T (v0 ) + 2b
u 0 Pb
v 0 T for all u0 , v0 ∈ V 0 , where P is some binary
symmetric m × m matrix. Note that Q0 has a subgroup L0 consisting of those
maps T + ε for which T is an additive map V 0 → Z4 . (Thus, the term ‘quadratic’
has to be viewed loosely.) Each P is associated with at least one form T = TP ,
by (4.1); each T determines a unique P (recall that P is a binary matrix); and,
by Lemma 4.3, the set of all T determining a given P is just the set of elements
of the coset TP + L0 that send 0 to 0.
Consider the quotient group Q0 /L0 (compare Cameron and Seidel [9]). Each of
its elements can be written as TP + L0 . If P and R are binary symmetric m × m
matrices, we claim that
(TP + L0 ) + (TR + L0 ) = TP⊕ R + L0 ,
(8.10)
where ⊕ again denotes binary addition. Namely, if a, b ∈ {0, 1} ⊂ Z4 then it is easy
to check that a + b = a ⊕ b + 2ab. By (4.1), if v0 ∈ V 0 and if b
v 0 = (α1 , . . . , αm ) is
0
any lift of v , then
X
X
TP (v0 ) + TR (v0 ) =
( Pjj + R jj )α2j + 2
( Pjk + R jk )α j αk
j <k
j
=
X
j
(P ⊕
R ) jj α2j
+2
X
j
Pjj R jj α2j + 2
X
j <k
( P ⊕ R ) jk α j αk ,
P
where b
v 0 7→ 2 j Pjj R jj α2j lies in L0 . Now (4.1) implies (8.10).
The Z4 -Kerdock code K4 (60 ) can be viewed as the set of functions in the
union of cosets TPA0 + L0 : this code is the set of |V 0 |-tuples of values of these
466
A . R . CALDERBANK
et al.
functions. Hence, K4 (60 ) is Z4 -linear if and only if these cosets form a subgroup;
by (8.10), this occurs if and only if the set of symmetric matrices PA0 is closed
under binary addition.
In view of Dembowski [12, pp. 220, 237], it follows that
C OROLLARY 8.11. If K4 (60 ) is Z4 -linear then the affine plane A(60 ) can be
coordinatized by a ( possibly non-associative ) division algebra.
The preceding proposition also suggests that we define
P4 (60 ) := K4 (60 )⊥
if K4 (60 ) is linear,
(8.12)
0
and call P4 (6 ) a Z4 -linear Preparata code: it has the same Lee weight
distribution as the Z4 -linear Preparata code constructed in [14]. Moreover, again
as in [14], φ(P4 (60 )) is a binary ‘Preparata’ code: it has the same (Hamming)
weight distribution as one of the original Preparata codes. We will study these
codes in detail in x 10.
9. Examples
0
E XAMPLE 9.1. The spread 6 that produces the desarguesian affine plane
A(6) = AG(2, 2m ) consists of the 1-spaces of GF(2m )2 : the line x = 0, and the
lines y = ax for a ∈ GF(2m ) (compare (5.2)). This spread is symplectic with
respect to any non-singular alternating bilinear form ( , ) on W = GF(2m )2 . If
Tr: GF(2m ) → GF(2 ) is the trace map, then Tr ( , ) is a non-singular alternating
bilinear form on the GF(2 )-space W . Since the members of 60 were totally
isotropic over GF(2m ), they remain totally isotropic over GF(2 ). Hence, 60
is a symplectic spread when W is viewed either as a GF(2m )-space or as a
m
GF(2 )-space. By Theorem 5.6, we obtain a family of 2m (2m + 1 ) lines in C2 .
E XAMPLE 9.2. Assume that m is odd. The (desarguesian) symplectic spread
described above determines an orthogonal spread 6 of an + (2m + 2, 2 )-space
via the geometric correspondence (7.2). The Kerdock set { M A | A ∈ 6 \ {Y (V )}}
in x 3 can be described as follows. Let m > 1 be odd, let L = GF (2rm ), let
K = GF(2r ), and let Tr: L → K be the trace map. For a ∈ L let Ma be the
K -linear map L ⊕ K → L ⊕ K given by
(x, α) Ma = (a2 x + a Tr(ax ) + αa, Tr(α x )) .
Then { Ma | a ∈ L } is a Kerdock set: if L ⊕ K is equipped with the bilinear
form (x, α) · ( y, β) := Tr(xy ) + αβ, then (x, α) Ma · (x, α) = 0 for all a, x, α, and
Ma + Mb is non-singular for any a 6= b in L . This is the ‘standard’ Kerdock
set when K = GF(2 ), producing the original Kerdock code K(6) (Dillon [13],
Kantor [16]). (Here we have switched to linear transformations L → L in place
of the mr × mr matrices appearing in the definition of Kerdock sets.)
On the other hand, in this case the Z4 -Kerdock code K4 (60 ) constructed in x 5
is the Z4 -linear code that appears in a paper by Hammons et al. [14, x 4.3]. The
construction of the Z4 -linear Kerdock and ‘Preparata’ codes by Hammons et al.
[14] requires the Galois ring GR (4m ), an extension of Z4 of degree m, containing
a (2m − 1 )th root of unity. To begin, let h2 (x ) ∈ Z2 [ X ] be a primitive irreducible
467
Z4 - KERDOCK CODES
polynomial of degree m. Then there is a unique monic polynomial h ( X ) ∈ Z4 [ X ]
m
of degree m such that h ( X ) ≡ h2 ( X ) (mod 2 ), and h ( X ) divides X 2 −1 − 1
m
(mod 4 ). Let ξ be a root of h ( X ), so that ξ2 −1 = 1. The Galois ring GR(4m ) is
defined to be Z4 [ξ]. Every element c ∈ GR (4m ) has a unique 2-adic representation
m
c = a + 2b, where a and b are taken from the set T = {0, 1, ξ, . . . , ξ2 −2 }. The
Frobenius map f from GR (4m ) to itself is the ring automorphism that takes any
element c = a + 2b ∈ GR (4m ) to c f = a2 + 2b2 . This map f generates the Galois
group of GR (4m ) over Z4 , and f m = 1. The relative trace from GR(4m ) to Z4 is
defined by
T (c ) = c + c f + . . . + c f
m −1
,
where c ∈ GR (4m ).
The Z4 -linear Kerdock code K and ‘Preparata’ code P constructed by Hammons
et al. [14] are
K = {(T (λ x ) + ε)x∈T | λ ∈ GR (4m ), ε ∈ Z4 }
and its dual code P = K⊥ . Thus, P is the Z4 -linear code consisting of the vectors
(c x )x∈T , with c x ∈ Z4 , such that
X
X
c x = 0 and
c x x = 0.
x∈T
x∈T
In order to connect the Kerdock code K with (5.3), write
K = {(T (λ1 x ) + 2T (λ2 x ) + ε)x∈T | λ1 , λ2 ∈ T, ε ∈ Z4 }.
Let ψ denote reduction modulo 2. Then ψ maps the Galois ring GR(4m ) to
the finite field GF(2m ). We will view V 0 as GF(2m ), and also as T: for each
v ∈ V 0 let ψ−1 (v) denote the preimage of v lying in T (this is our old b
v).
The map v 7→ 2T [λ1ψ−1 (v)] is a linear functional on GF (2m ). We claim that
v 7→ T [λ1ψ−1 (v)] defines a Z4 -valued quadratic form on GF (2m ):
T [λ1ψ−1 (v + w)] = T [λ1ψ−1 (v)] + T [λ1ψ−1 (w)] + 2 B (v, w)
for some symmetric bilinear form B ( , ) on V 0. In order to see this, write
ψ−1 (v) + ψ−1 (w) = a + 2b with a, b ∈ T, so that a = ψ−1 (v + w). Then
a = (a + 2b )2 = (ψ−1 (v) + ψ−1 (w))2
= ψ−1 (v) + ψ−1 (w) + 2[ψ−1 (v)ψ−1 (w)]1/2 ,
m
m
so that
T [λ1ψ−1 (v + w)] − T [λ1ψ−1 (v)] − T [λ1ψ−1 (w)] = 2T [λ21ψ−1 (v)ψ−1 (w)],
as required. Consequently, K can, indeed, be written as in (5.3).
E XAMPLE 9.3. When m is odd and composite, Kantor [17] constructed exponential numbers of inequivalent orthogonal spreads of + (2m + 2, 2 )-space
(‘inequivalent’ means that these are in different O+ (2m + 2, 2 )-orbits). These
spreads produce Kerdock sets and Kerdock codes over both Z2 and Z4 . However,
by Proposition 8.9 and Corollary 8.11 the Z4 -codes are linear if and only if the
associated set of symmetric matrices is closed under addition, in which case the
associated translation plane A(60 ) is coordinatized by a non-associative division
algebra. Very few examples of such spreads are known; they can all be described
as follows [16].
468
A . R . CALDERBANK
et al.
Let n > 1 be odd, let L = GF(2rn ), let K = GF(2r ), and let Tr: L → K be the
trace map. For a ∈ L let Pa be the K -linear map L → L given by
x Pa = a2 x + a Tr(x ) + Tr(ax ).
Then a 7→ Pa is additive, and, if a 6= 0, then Pa is non-singular (Kantor [16, II,
(5.4)]). (Proof. If x Pa = 0, write z = ax. Then z2 + z Tr(x ) + x Tr (z ) = 0. Apply
Tr and obtain Tr(z )2 + Tr(z ) Tr (x ) + Tr (x ) Tr (z ) = 0. Then Tr(z ) = 0 and hence
z2 + z Tr(x ) = 0. If z 6= 0 then z = Tr (x ) ∈ K ; but then Tr (z ) = z since n is odd,
which produces the contradiction z = 0 since Tr (z ) = 0.)
Now equip L with the non-singular symmetric bilinear form Tr(xy ), for
x, y ∈ L . Then a straightforward calculation shows that
Tr ((x Pa ) y ) = Tr (x ( y Pa )) for all a, x, y ∈ L ,
so that Pa is symmetric (with respect to an orthonormal basis of L ). Passing
to GF(2 ) by using the trace map K → GF(2 ) as in Example 9.1, we obtain a
non-singular symmetric bilinear form on the GF(2 )-space L with respect to which
each Pa is symmetric.
When m = rn is odd, let 60 (r, n ) denote the symplectic spread of F obtained
using this GF(2 )-space of symmetric matrices, and let 6(r, n ) denote the
corresponding orthogonal spread of E obtained by means of (7.2). The following
proposition summarizes information contained in a paper by Kantor [16].
P ROPOSITION 9.4. (i) The spreads 60 (r1 , n1 ) and 60 (r2 , n2 ) are inequivalent if
(r1 , n1 ) 6= (r2 , n2 ).
(ii) The spread 6(1, m ) is equivalent to the orthogonal spread in Example
9.2, but 60 (1, m ) is not equivalent to the desarguesian spread in Example 9.1 if
m > 3.
(iii) If 60 is a symplectic spread of F that behaves as in Proposition 8.9
and whose associated orthogonal spread (cf. (7.2 )) is equivalent to the one in
Example 9.2, then 60 is equivalent to either the desarguesian spread or 60 (1, m ).
(iv) The spread 6(r, n ) is not equivalent to 6(1, rn ) if r > 1.
Proof. (i) [16, II, (5.5),(7.1)].
(ii) [16, I, (4.1) and II, x 5].
(iii) [16, I, (4.1)].
(iv) This follows from (iii).
In general, 6(r1 , n1 ) and 6(r2 , n2 ) are inequivalent if (r1 , n1 ) 6= (r2 , n2 ). This
and related questions have been addressed by Michael Williams in his thesis [34].
10. Inequivalence
We now return to the general framework of x 5, with the goal of discussing
the problem of equivalence among the Gray images of the Z4 -linear Preparata
codes described above and those binary Preparata codes known previously. We
begin by writing a generator matrix G4 (60 ) for the Z4 -linear Kerdock code
K4 (60 ) given by (5.3). As in x 8, the 2m coordinate positions are labelled by the
469
Z4 - KERDOCK CODES
vectors v in V 0 = Z2m . Let P1 , . . . , Pm be a basis for the space h PA | A¯ ∈ 60 i, so
that the Z4 -valued quadratic forms Ti = TPi , for 1 < i < m, span the Z4 -module
hTPA | A¯ ∈ 60 i. Then


1 ···
1
··· 1
· · · T1 (v) · · ·


.
G4 (60 ) = 
..


.
···
Tm (v)
···
The Kerdock code K constructed by Hammons et al. [14] is generated by the
matrix


1 ···
1
··· 1


···
···
T (x )


···
···
T (ζ x )

,


..


.
· · · T (ζ m−1 x ) · · ·
where the 2m coordinate positions are indexed by T = {0, 1, ζ, . . . , ζ 2 −2 } and
T has been identified with V 0 as in Example 9.2. Recall that T (ζ i x ) defines a
Z4 -valued quadratic from, as observed above in Example 9.2.
Note that h2TPA | A¯ ∈ 60 i is the binary vector space of all linear functionals
v 7→ 2b
v , where s ∈ V 0. (Hence, T1 , . . . , Tm is a Z4 -basis of the Z4 -module
s ·b
hTPA | A¯ ∈ 60 i.) Write si = d ( Pi ) (cf. (4.1) and Lemma 7.3), so that s1 , . . . , sm is
a basis of V 0, and the matrix


1 ···
1
··· 1


· · · 2b
v ···
s1 · b

G4 = 
..


.
m
···
2b
v ···
sm · b
generates a Z4 -linear code RM(1, m )4 that is common to all of the Kerdock
codes K4 (60 ) given by (5.3). (This notation RM(1, m )4 arises from the fact [14,
Theorem 7] that its Gray image φ(RM(1, m )4 ) is the usual Z2 -linear first-order
Reed–Muller code RM(1, m + 1 ) of length 2m+1 .)
The matrix G4 (60 ) is a parity check matrix for the Z4 -linear Preparata code
P4 (60 ) defined in (8.12). Also, G4 is a parity check matrix for a Z4 -linear code
H4 = RM(1, m )4⊥ . We note in passing that the Lee weight distribution of H4 is the
MacWilliams transform of that of RM(1, m )4 , and hence the weight distribution
of the binary code φ( H4 ) is the MacWilliams transform of that of RM(1, m + 1 ):
the weight distribution of φ( H4 ) is the same as that of the Z2 -linear extended
Hamming code of length 2m+1. What is important for us is the fact that, when
m > 5, the binary code φ( H4 ) is non-linear [14, Theorem 8]. (When m = 3,
φ( H4 ) is the Hamming code [14, Theorem 18].)
Since RM(1, m )4 ⊆ K4 (60 ), we have P4 (60 ) = K4 (60 )⊥ ⊆ RM(1, m )4⊥ = H4 .
e4 denote the Z4 -linear code whose parity check matrix is
Let H

 

2 ···
2
2 ···
2
··· 2
··· 2
· · · 2T1 (v) · · ·
· · · 2b
v ···
s1 · b

 

=
.
2G4 (60 ) = 
..
..




.
.
· · · 2b
v ···
sm · b
· · · 2Tm (v) · · ·
470
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et al.
e4 , and φ( H
e4 ) is Z2 -linear.
Then H4 ⊆ H
e4 ) is the binary span of φ( H4 ). The codewords
L EMMA 10.1. For m > 5, φ( H
of weight 2 in φ( H4 ) are the vectors vi + v2m +i , with i = 1, . . . , 2m , where
m+1
v1 , . . . , v2m+1 is the standard basis of V = Z22 .
e4 and φ( H
e4 ) is linear, hφ( H4 )i ⊆ φ( H
e4 ). Moreover, since
Proof. Since H4 ⊆ H
φ( H4 ) is non-linear,
m
22
−m −2
e4 )| = 22
= |φ( H4 )| < |hφ( H4 )i| < |φ( H
m
−m −1
,
e4 ).
and it follows that hφ( H4 )i = φ( H
Thus, we must determine the words of weight 2 in φ( H4 ). The words of Lee
weight 2 in Z4 are those whose coordinates consist either of a single 2 and all
others 0, or of two ±1’s and all others 0. The first type of word annihilates G4 ,
and its Gray image is exactly a vector vi + v2m +i for some i. No word of the
second type can lie in H4 since linear functionals separate points.
Zaitzev, Zinoviev and Semakov [36] proved that any binary code P of length
2m+1 with the same Hamming distance distribution as the original Preparata code
[29] is contained in a uniquely determined binary code H with the same Hamming
distance distribution as the extended Hamming code of length 2m+1 . Moreover, H
is the union of P together
S with all binary vectors at Hamming distance 4 from P
(or equivalently, H = { P + w | w has weight 4}). Note that the Gray map can
m
be used to translate this theorem into the metric space (Z24 , Lee metric); this
was done explicitly by Hammons et al. [14, x 5.4] in the case of the Preparata
code P4 (60 ), where 60 is the desarguesian spread. In the case of the ‘standard
binary Preparata codes’, the above partition was studied by Preparata [29], Baker
et al. [3] and Kantor [18].
L EMMA 10.2. For m > 5, no standard binary Preparata code is equivalent to
the Gray image φ(P4 (60 )) of a Z4 -linear Preparata code P4 (60 ).
Proof. Any equivalence would be a linear transformation of the underlying
m+1
vector space Z22 , and hence send subspaces to subspaces. The binary span of
any standard Preparata code is the usual extended Hamming code. However, we
have already observed that φ(P4 (60 )) spans a larger code (by Lemma 10.1, it
has words of weight 2).
T HEOREM 10.3. Let m > 5, and let K4 and K]4 be two Z4 -linear Kerdock
]
]⊥
codes arising as in (5.3 ). Let P4 = K⊥
be the corresponding
4 and P4 = K4
Z4 -linear Preparata codes. Then the following are equivalent:
(i) K4 and K]4 are equivalent linear codes in Z24 ;
m
(ii) P4 and P]4 are equivalent linear codes in Z24 ;
m
(iii) φ(K4 ) and φ(K]4 ) are equivalent codes in Z22
m+1
(iv) φ(P4 ) and
φ(P]4 )
are equivalent codes in
; and
m+1
Z22 .
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Z4 - KERDOCK CODES
m
Proof. Since monomial transformations of Z24 preserve the inner product, (i)
and (ii) are equivalent. Before discussing (iv) we need to discuss the monomial
m+1
m
transformations of Z22
induced from those of Z24 . If g is any monomial
m
m+1
transformation of Z24 , then an isometry h of Z22 is determined by the following
diagram:
g
m
m
Z24
Z24
6
φ −1
m+1
Z22
-
φ
?
hZ
2m+1
2
Then h is a permutation of coordinates. This already shows that (i) implies (iii)
and (ii) implies (iv), but we will need to be more explicit. The following two
diagrams show the effect of h on the entries in the ith and (i + 2m )th coordinate
positions for i = 1, . . . , 2m ; the position j depends upon the effect of the monomial
transformation g, as does the effect on the entry a + 2b in position i:
ith
a + 2b
φ
−1
g
6
- a + 2b
jth
φ
(b | a + b )
ith (i+2m )th
h
- (b | a?+ b )
jth ( j+2m )th
ith
a + 2b
φ
−1
g
6
(b | a + b )
ith (i+2m )th
- a + 2(a + b )
jth
= −(a + 2b )
φ
h
- (a + b?| b )
jth ( j+2m )th
In particular, it follows that the permutation h preserves the partition π =
{{i, i + 2m } | i = 1, . . . , 2m }}. Conversely, every permutation of the coordinates of
+1
that preserves π arises from some monomial transformation of Zm
Zm
4 . This
2
can be seen either from the above diagrams, or by observing that the group of
monomial permutations of Zm
of permutations of the coordinates
4 , and the group
+1
2m m
of Zm
preserving
,
both
have
order
2
2
!.
π
2
We are now ready to prove that (iv) implies (ii) in the theorem. As noted prior
to the proof of Lemma 10.2, φ( H4 ) is uniquely determined by φ(P4 ), and hence
so is its linear span hφ( H4 )i as well as the set of vectors of weight 2 in that span.
Consequently, by Lemma 10.1 the partition π can be recovered from φ(P4 ), and
+1
hence so can both φ and P4 . If h is a coordinate permutation of Zm
that sends
2
]
φ(P4 ) to φ(P4 ), it follows that h preserves π, and hence lifts to an equivalence
P4 → P]4 .
In order to prove that (iii) implies (i), we will study combinatorial aspects of
the binary Kerdock code K := φ(K4 ) in a manner superficially similar to what
we have just done in order to prove that (iv) implies (ii). There is an equivalence
relation on K defined as follows: two words are equivalent if and only if they
are at distance 0, 2m or 2m+1 ; the equivalence classes are precisely the cosets of
RM(1, m + 1 ) (cf. Proposition 3.12). We will use the description of K coming
from the proof of Theorem 8.3: each coset looks like
J A := (v0 U PA v0T | v0 U PA v0 T + d ( PA ) · v0 ) + RM(1, m + 1 )
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A . R . CALDERBANK
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for some A ∈ 6\{Y (V )}; here, PA is as in (5.1)–(5.3), and U PA is the ‘upper
triangular’ portion of PA + d ( PA )T d ( PA ) as in (8.7), so that PA + d ( PA )T d ( PA ) =
U PA + U PTA .
Let J denote the following subset of Z2V :
J := {a + b + c | a ∈ J A , b ∈ J B , c ∈ JC , with A , B , C ∈ 6\{Y (V )}
and PA + PB + PC = 0} + RM(1, m + 1 ).
If PA + PB + PC = 0 then d ( PA ) + d ( PB ) + d ( PC ) = 0. Thus, each coset of
RM(1, m + 1 ) contained in J has a coset representative that looks like
(v0 (U PA + U PB + U PC )v0 T | v0 (U PA + U PB + U PC )v0 T ).
By the definition of U PA , U PB , U PC ,
(U PA + U PB + U PC ) + (U PA + U PB + U PC )T
= d ( PA + PB )T d ( PA + PB ) + d ( PA )T d ( PA ) + d ( PB )T d ( PB )
= d ( PA )T d ( PB ) + d ( PB )T d ( PA ).
Thus, U PA + U PB + U PC + d ( PA )T d ( PB ) is symmetric.
It follows that
v0 (U PA + U PB + U PC )v0 T = v0 d ( PA )T d ( PB )v0 T + v0 d (d ( PA )T d ( PB ))v0 T
= [d ( PA ) · v0 ][d ( PB ) · v0 ] + d (d ( PA )T d ( PB )) · v0 .
Then
J = {([d ( PA ) · v0 ][d ( PB ) · v0 ] | [d ( PA ) · v0 ][d ( PB ) · v0 ] ) | A , B ∈ 6\{Y (V )}}
+ RM(1, m + 1 ).
Since { M A | A ∈ 6\{Y (V )}} is a Kerdock set, d ( PA ) can be any vector in V 0 (cf.
(7.4)). Thus,
J = { (`1 (v0 )`2 (v0 ) + µ | `1 (v0 )`2 (v0 ) + µ + λ) | `1 , `2 ∈ V 0∗ , µ, λ ∈ Z2 },
where V 0∗ denotes the dual space of V 0 (note that `1 (v0 )`1 (v0 ) = `1 (v0 )).
We now turn to the subspace J ⊥ of Z2V (of course, J itself is not a
subspace). For any distinct u 1 , u 2 ∈ V 0, let w(u 1 , u2 ) denote the word in Z2V whose
(u 1 0 ), (u 1 1 ), (u 2 0 ) and (u 2 1 ) coordinates are 1 while all others are 0. Then
w(u 1 , u 2 ) is a word of weight 4 in J ⊥ .
We claim that these are the only words of weight 4 in J ⊥ . For, suppose
that there is another word with this property. Then this must correspond to the
following equation holding for all `1 , `2 ∈ V 0∗ and some distinct u 1 , u 2 , u 3 ∈ V 0:
4
X
`1 (ui )`2 (ui ) = γ,
a constant.
1
It is easy to check that this is impossible. (Namely, we may assume that
u 1 6= 0, u 4 . Choosing `1 and `2 to vanish at all u i we see that γ = 0; choosing `1
and `2 so that `1 (u 1 ) = `2 (u 1 ) = 1 and `1 (u i )`2 (u i ) = 0 for i 6= 1, we obtain the
contradiction γ = 1.)
It is easy to check that these words w(u 1 , u 2 ) of weight 4 have the following
property: two vectors (v01 ε 1 ), (v02 ε 2 ) ∈ V appear in more than one word w(u 1 , u 2 )
if and only if v01 = v02 .
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Z4 - KERDOCK CODES
Now note that J and J ⊥ were obtained canonically from K, and hence so
is the partition π discussed earlier in this proof. This means that K4 has been
recovered from K. Moreover, as we saw earlier, it follows that any equivalence
φ(K4 ) → φ(K4] ) lifts to an equivalence K4 → K4] , as required.
The above proof also showed the following.
P ROPOSITION 10.4. If K4 is a Z4 -linear Kerdock code of length m > 5, then
there is only one Z4 -linear structure on φ(K4 ), and only one on φ(P4 ).
We have seen that, in the Z4 -linear case, each equivalence φ(K4 ) → φ(K4] )
lifts to an equivalence K4 → K4] . What does such a lifting look like in
terms of orthogonal spreads? We will consider this question in detail, in one
important case, without assuming Z4 -linearity. In order to state our result
precisely, we need to recall some notation from xx 2 and 5. As in earlier
sections, coordinate vectors and matrices will be written with respect to the basis
X (v1 ), . . . , X (vm+1 ), Y (v1 ), . . . , Y (vm+1 ) of E . For example, X (vm+1 ) is the vector
with 1 in position m + 1 and 0 elsewhere.
Given an orthogonal spread 6 of E , as usual we will consider the symplectic
spread 60 = 6ω¯ of F obtained from it as (7.1) using ω¯ = X (vm+1 )Y (vm+1 ).
Corresponding to 60 is a set S(60 ) of binary symmetric matrices obtained as
in (5.1). Earlier, we were able to avoid cumbersome notation involving the
dependence of all of these objects on choices made for our two members X (V )
and Y (V ) of 6 as well as on the non-singular point ω¯ , but now we will have to be
somewhat more careful. Therefore, we will write S(60 ) = S(6, X (V ), Y (V ), ω)
¯ ;
but note that we are still suppressing the notation for the dual bases chosen
for X (V ) and Y (V ). Similarly, we will write K4 (6, X (V ), Y (V ), ω)
¯ instead of
K4 (60 ). Now we can state the following general result concerning the equivalence
of Z4 -Kerdock codes.
T HEOREM 10.5. Let 6 and 6] be orthogonal spreads of E containing X (V )
and Y (V ). Let ω¯ be as above. Then the following are equivalent:
(i) K4 (6, X (V ), Y (V ), ω)
¯ and K4 (6] , X (V ), Y (V ), ω)
¯ are equivalent Z4 Kerdock codes;
(ii) there are an invertible m × m matrix R, and a vector s0 ∈ V 0, such
that the map P 7→ R[ P + d ( P )T s0 + s0T d ( P )] R T sends S(6, X (V ),
Y (V ), ω)
¯ to S(6] , X (V ), Y (V ), ω)
¯ ; and
(iii) there is an orthogonal transformation of E that sends 6 to 6] while
fixing X (V ), Y (V ), and X (vm+1 ).
m
Proof. Assume (i), and let g be a monomial transformation of Z24 that
induces an equivalence K4 (60 ) → K4 (60] ). By (4.1) or (4.2) together with (5.2),
the subcode 2K4 (60 ) can be identified with the first-order Reed–Muller code
(here, Z2 is identified with Z4 /2Z4 ). Since 2K4 (60 )g = 2K4 (60] ), the equivalence
g permutes coordinates via an affine transformation v0 7→ v0 R + w0 for some
non-singular m × m matrix R and some w0 ∈ V 0.
Since g is a monomial transformation, it sends the all-one word to a word in
K4 (60] ) all of whose coordinates are ±1. In the notation of (5.3), this means that,
474
A . R . CALDERBANK
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for some A , TPA (v0 ) + ε = ±1 for all v0 ∈ V 0. By (8.6), d ( PM ) · b
v 0 + ε = ±1 for
0
0
all v0 ∈ V 0, and hence 0 + ε = ±1 and d ( PM ) · b
0
for
all
v =
v ∈ V 0. Consequently,
g sends the all-one word to one of the form
v0 + 1 )v0 = ±((−1 )s ·v )v0
s0 · b
±(2b
0
(10.6)
with s0 ∈ V 0 . (Note. This is a remarkable identity associated with the ring Z4 : it
converts between additive and multiplicative notation in an unexpectedly simple
manner.) It follows that g is given by
0
0
0
(cv0 )g = (−1 )s ·(v R +w ) cv0 R +w0
for some s0 ∈ V 0.
By (5.3), the translation v0 7→ v0 + w0 is an automorphism of K4 (60] ). Hence,
we may suppose that the equivalence g takes the form
0 0
(cv0 )g = (−1 )s ·v R cv0 R .
Let (TP (v0 ))v0 ∈ K4 (60 ) arise from the Z4 -valued quadratic form TP corresponding (as in (4.2)) to the binary symmetric matrix P ∈ S(60 ), and let (TP (v0 ))v0 g =
(T ] (v0 ))v0 ∈ K4 (60] ). We need to calculate 1 := T ] (u0 + v0 ) − T ] (u0 ) − T ] (v0 ) in
order to determine the binary symmetric matrix underlying the Z4 -valued quadratic
form T ] (v0 ) as in (4.2):
0
0
0
0
0
1 = (−1 )s ·(u R +v R ) TP (u0 R + v0 R ) − (−1 )s ·(u R ) TP (u 0 R )
0
0
− (−1 )s ·(v R ) TP (v0 R )
0
0
0
0 R Pv
0 RT ]
d
= (−1 )s ·(u R +v R ) [TP (u0 R ) + TP (u 0 R ) + 2ud
0
0
0
0
− (−1 )s ·(u R ) TP (u0 R ) − (−1 )s ·(v R ) TP (v0 R )
0
0
0
0
0
= TP (u0 R )[ (−1 )s ·(u R +v R ) − (−1 )s ·(u R ) ]
0
0
0
0
0
0 R Pv
0 RT
d
+ TP (v0 R )[ (−1 )s ·(u R +v R ) − (−1 )s ·(v R ) ] + 2ud
0 R + T (v0 R )2b
0 R + 2u
0 R Pv
0 RT ,
d
d
s0 · vd
s0 · ud
= TP (u0 R )2b
P
0 R, while 2[dd
0 R][b
0 R] =
using (10.6). By (4.1), 2TP (v0 R ) = 2dd
( P ) · ud
( P ) · ud
s0 · ud
T 0 0 T
0
d
ud
Rd ( P ) b
s vd
R by straightforward matrix multiplication. Thus,
T 0
T
0 R[ P + dd
0 RT
1 = 2ud
( P) b
( P )]vd
s +b
s0 dd
0 R[ P + d ( P )T b
( P )] R T b
v0 .
s0 + b
s0 dd
= 2ud
T
T
It follows from (4.2) that the binary symmetric matrix associated with (TP (v0 ))v0 g
is R[ P + s0 T d ( P ) + d ( P ) T s0 ] R T . This is precisely the assertion in (ii).
Conversely, starting with S(6, X (V ), Y (V ), ω)
¯ , S(6] , X (V ), Y (V ), ω)
¯ , R and
0 as in (ii), reverse the preceding argument in order to obtain (i).
s
Next we show that (ii) implies (iii). Let P ∈ S(6, X (V ), Y (V ), ω),
and
¯
write P] = RT [ P + s0 T d ( P ) + d ( P ) T s0 ] R, so that P] ∈ S(6] , X (V ), Y (V ), ω)
¯ by
hypothesis. Let M and M ] be the skew-symmetric matrices obtained, respectively,
from P and P] using (7.4). Then
]
P + d ( P] ) T d ( P] ) d ( P] ) T
M1 d ( P )T
and M ] =
,
M=
0
0
d ( P] )
d ( P)
475
Z4 - KERDOCK CODES
where M1 = P + d ( P )T d ( P ). Let
B=
R− T
O
sT
1
and
e=
B
B
O
O
−T .
B
(10.7)
Then e
B is the matrix of an element of O+ ( E ) (cf. Lemma 2.9(i)), and


I
O R[ M1 + s0 T d ( P ) + d ( P ) T s0 ] R T R T d ( P )T
1
0
d ( P) R
O

e−1 I M e
B=
B
.
O
O
I
O
O I
0
1
O
O
Here,
R T [ M1 + s0 T d ( P ) + d ( P ) T s0 ] R
T
= R T [ P + s 0 d ( P ) + d ( P ) T s 0 ] R + [d ( P ) R ] T [d ( P ) R ] ,
where d ( R T PR ) = d ( P ) R = d P + s0 T d ( P ) + d ( P ) T s 0 R since
X
Rki Pkj R ji
( R T PR )ii =
kj
=
X
( Rki Pkj R ji + R ji P jk Rki ) +
j <k
=
X
X
Rki Pkk Rki
k
Pkk Rki
k
(as Rki Pkj R ji = R ji P jk Rki ). Consequently,
]
e−1 I M B
e= I M ,
B
O
I
O I
e sends 6 to 6] since it fixes Y and
and B
]
I M e
e−1 I M B
e= X I M .
B = XB
X
O
I
O I
O I
e behaves as in (iii).
Thus, B
e is a matrix inducing an orthogonal transformation
Conversely, suppose that B
e can be written as in (10.7) for some invertible matrix R and
as in (iii). Then B
some s0 ∈ V 0 . Reversing the previous argument, we find that
R T [ P + s0 T d ( P ) + d ( P ) T s0 ] R ∈ S(6] , X (V ), Y (V ), ω)
¯
whenever P ∈ S(6, X (V ), Y (V ), ω)
¯ , and hence that (ii) holds.
C OROLLARY 10.8. If K4 (6, X (V ), Y (V ), ω)
¯ and K4 (6] , X (V ), Y (V ), ω)
¯ are
two equivalent Z4 -Kerdock codes, then the binary Kerdock codes K(6) and
K(6] ) are also equivalent.
C OROLLARY 10.9. For each odd composite integer m, there are more √
than
√
m
2 m/2 inequivalent Z4 -Kerdock codes of length 22 , and more than 2 m/2
m+1
inequivalent Z2 -Kerdock codes of length 22 .
Proof. By a result of Kantor [18], there are more than 2
orthogonal spreads of E .
√
m/2
inequivalent
476
A . R . CALDERBANK
et al.
The codes in the previous corollary are non-linear. The argument in the next
consequence of Theorem 10.5 explains what condition (iii) means when dealing
with the case 6 = 6] .
C OROLLARY 10.10. The Z4 -linear Kerdock codes arising from Examples 9.2
and 9.3 are equivalent.
Proof. The desarguesian spread 60 of F in Example 9.1 yields an orthogonal
spread 6 of E as in Example 9.2, and X (V ), Y (V ) ∈ 6. Fix s0 ∈ V 0 \ {0},
e be as in (10.7), and set 6] := 6 B
e. Then B
e
write R = I , let B and B
induces an orthogonal transformation of E having order 2 and behaving as in
Theorem 10.5. It remains to describe the Z4 -Kerdock codes K4 (6, X (V ), Y (V ), ω)
¯
and K4 (6] , X (V ), Y (V ), ω)
¯ .
The code K4 (6, X (V ), Y (V ), ω)
¯ is just the Z4 -Kerdock code in Example 9.2:
the one discovered by Hammons et al. [14].
The code K4 (6] , X (V ), Y (V ), ω)
¯ , the set S(6] , X (V ), Y (V ), ω)
¯ of symmetric
matrices, and a certain symplectic spread determine one another as in (5.8) and
(5.3). This symplectic spread is the one called 6] ω¯ in (7.1). It is equivalent
e)ω¯ Be = 6ω¯ e
e = X (vm+1 )Y (s0 + vm+1 ). By
to the symplectic spread (6] B
¯B
B . Here, ω
0
Proposition 9.4(ii), if s 6= 0 then 6ω¯ e
B is equivalent to the symplectic spread in
Example 9.3.
C OROLLARY 10.11. If m is odd and composite then there is a Z4 -linear
Kerdock code not equivalent to the one in Example 9.2. The corresponding
Z4 -linear Preparata code and binary Preparata code are also not equivalent to
ones arising from Examples 9.2 and 9.3.
Proof. By Proposition 9.4(iii), if m = rn with r, n > 1, then the orthogonal spreads 6(r, n ) and 6(1, m ) are not equivalent. Hence, neither are the
Kerdock codes K4 (6(r, n ), X (V ), Y (V ), ω)
¯ and K4 (6(1, m ), X (V ), Y (V ), ω)
¯ , by
Theorem 10.5. The case of Preparata codes follows from Theorem 10.3.
A census of binary Preparata codes
The following binary Preparata codes are presently known.
(i) The original Preparata codes (Preparata [29]).
(ii) The variants of (i) studied by Baker et al. [3] and Kantor [18].
(iii) The codes (other than the one of length 24 in (i)) constructed by
Hammons et al. [14] as Gray images of Z4 -linear codes.
(iv) The codes in Corollary 10.11.
The isomorphisms among the codes (i) and (ii) were completely determined in
[18]. These codes span the usual extended Hamming code; the codes in (iii) and
(iv) do not, by Lemma 10.1. Consequently, the codes in (iii) and (iv) are not
equivalent to any in (i) or (ii), and they are not equivalent to one another by
Proposition 9.4.
Additional Z4 -linear Preparata codes can be constructed by using Example 9.3
together with (7.4). Very recently, large numbers of other binary and Z4 -linear
477
Z4 - KERDOCK CODES
Preparata codes have been obtained by a generalization of the constructions in
Example 9.3 and the preceding corollaries [34].
11. Extraspecial p-groups, symplectic spreads and complex line-sets
with prescribed angles
Let q = pm for any odd prime p and any integer m > 1. Let V be an
m-dimensional vector space over GF( p ), and label the standard basis of C q as ev ,
where v ∈ V . Equip C q with the usual hermitian inner product; again let U (Cq )
be the corresponding unitary group. Let ξ be a primitive p th root of unity in C.
For b ∈ V define elements of U (Cq ) by
X (b ) : ev 7→ ev+b
and
Y (b ) := diag[ξb·v ].
Let X (V ) := { X (b ) | b ∈ V } and Y (V ) := {Y (b ) | b ∈ V }.
L EMMA 11.1. The group E := h X (V ), Y (V )i is extraspecial of order p1+2m ,
and Z ( E ) = hξ I i. Moreover, E = X (V )Y (V )h− I i, and every element E can be
written uniquely in the form X (a )Y (b )(ξ I )γ for some a, b ∈ V and γ ∈ GF( p ).
Again let E = E / Z ( E ), and for b ∈ V write e∗b = p−m/2
P
vξ
b·v
ev .
L EMMA 11.2. The following hold:
(i) {hev i | v ∈ V } is the set of irreducible submodules for Y (V );
(ii) {he∗b i | b ∈ V } is the set of irreducible submodules for X (V ).
We identify Z ( E ) with GF( p ) in order to obtain a non-singular alternating bilinear form on E defined by (e 1 , e 2 ) = [e1 , e2 ] ∈ Z ( E ). As in (2.5),
X (a )Y (b ), X (a0 )Y (b0 ) = a0 · b − a · b 0 . The m-spaces X (V ) and Y (V ) are totally
isotropic and have only 0 in common.
There is a subgroup GL(V ) of U (C q ) normalizing E and acting on E exactly
as in Lemma 2.9(i).
L EMMA 11.3. Let A and B be totally isotropic m-spaces of the symplectic
space E over GF( p ).
(i) The set F( A ) of A-irreducible subspaces of C q is an orthogonal frame.
(ii) If A ∩ B = 0, and if u1 and u2 are unit vectors in different members of
F( A ) ∪ F( B ), then |(u1 , u2 )| = 0 or p−m/2.
T HEOREM 11.4. Let 6 be any symplectic spread of the symplectic space E.
Then
[
F(6) :=
F( A )
A∈6
consists of q(q + 1 ) lines with the following property: if u1 and u2 are unit
vectors in different members of F(6) then |(u1 , u2 )| = 0 or p−m/2.
The complex line-set F(6) is extremal in the sense described following (5.9).
Some of these line-sets (those corresponding to desarguesian planes) appear in
478
A . R . CALDERBANK
et al.
papers by Levenštein [25, p. 529] and König [22, Theorem 2] stripped of our
group-theoretic and geometric contexts.
L EMMA 11.5. The group of all elements of U (C q ) sending each of the frames
F( X (V )), F(Y (V )), to itself normalizes E.
C OROLLARY 11.6. Let 61 and 62 be symplectic spreads of E.
(i) Any symplectic transformation sending 61 to 62 is induced by an element of
U (C q ) sending F(61 ) to F(62 ) (in fact by many such unitary transformations ).
In particular, the set-stabilizer of 61 in Sp(2m, p ) is induced by a subgroup of
m
U (C p ) preserving the frame F(6).
(ii) The pointwise stabilizer of F(6) in U (C q ) is h E , C I i where C ⊆ C∗ is the
unit circle.
(iii) Any element of U (C q ) sending F(61 ) to F(62 ) induces a projective
symplectic transformation of E sending 61 to 62 .
Note that (4.5) and Lemma 4.7 hold without change with d M := diag[ξv M v /2 ].
As in Example 1 of x 9, desarguesian planes produce examples of symplectic
spreads and hence of extremal complex line-sets. However, there are relatively
few examples known of non-desarguesian symplectic spreads in odd characteristic.
These are surveyed in [2] and [19]. Once again, in spite of the small numbers
known, there must be large numbers of inequivalent examples.
T
The odd behaviour of 4
It is natural to wonder, both in the context of Hammons et al. [14] and the
present paper, whether there might be a theory of codes over rings other than Z4
paralleling these recent developments. While there is no way to ‘prove’ that no
such analogues exist, our geometric and group-theoretic points of view provide
some negative indications.
First of all, this section dealt with symplectic spaces, not orthogonal ones. There
is no analogue in odd characteristic of the peculiar but wonderful relationships
between orthogonal and symplectic spaces and groups. In characteristic 2,
every quadratic form produces an alternating bilinear form (since (v, v) =
Q (v + v) − Q (v) − Q (v) = 0). Nothing comparable occurs in odd characteristic,
neither from a purely computational point of view nor within finite group theory
or finite geometry.
Next, in x 4 we extended an extraspecial 2-group E m to a slightly larger group
F . This had the effect of increasing the numbers of elements of order 2 and 4
without introducing elements of larger orders. This is what produces an action of
a full symplectic group, and helps to explain the occurrence of this slightly larger
group F . Extending further, by further increasing the size of the centre, would
also introduce elements of larger orders. Such extensions are of less interest, and
do not arise in any natural group-theoretic context (compare [1, pp. 109–111]).
This proves nothing, it merely suggests that neither GF( p ) with p odd, nor
Z8 , is likely to produce results close to what has been discovered using Z4 . The
natural progression on the right-hand side of our ‘road map’ in Table 1 is to
the quaternion world. This suggests that the ‘correct’ generalization will involve
Z4 - KERDOCK CODES
479
a non-abelian group in place of Z4 ; the quaternion case will be investigated
elsewhere.
Acknowledgement. We are grateful to J. I. Hall and a referee for numerous
helpful comments.
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A. R. Calderbank
AT&T Bell Laboratories
Murray Hill
New Jersey 07974
U.S.A.
P. J. Cameron
School of Mathematical Sciences
Queen Mary and Westfield College
London E1 4NS
W. M. Kantor
Department of Mathematics
University of Oregon
Eugene
Oregon 97403
U.S.A.
J. J. Seidel
Faculty of Mathematics and
Computing Science
Eindhoven University of Technology
5600 MB Eindhoven
The Netherlands