IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
367
Consta-Abelian Codes Over Galois Rings
Kiran. T and B. Sundar Rajan, Senior Member, IEEE
Abstract—We study -length consta-Abelian codes (a generalization of
the well-known Abelian codes and constacyclic codes) over Galois rings of
characteristic , where and are coprime. A twisted discrete Fourier
transform (DFT) is used to generalize transform domain results of Abelian
and constacyclic codes, to consta-Abelian codes. Further, we characterize
consta-Abelian codes invariant under two kinds of monomials, whose underlying permutations are effected by: i) multiplying the coordinates with
a unit in the appropriate mixed–radix representation of the coordinate positions and ii) shifting the coordinates by positions. All the codes studied
here belong to the class of quasi-twisted codes which are known to contain
some good codes. We show that the dual of a consta-Abelian code invariant
under the two monomials is also a consta-Abelian code closed under both
monomials.
Index Terms—Abelian codes, consta-Abelian codes, constacyclic codes,
Galois rings, quasi-twisted (QT) codes, twisted discrete Fourier transform.
Fig. 1. Generalizations of cyclic code.
I. INTRODUCTION AND PRELIMINARIES
The class of cyclic codes and its generalizations have been extensively studied, of which the following are the most important.
• Abelian codes: These are ideals in the group algebra Fq [G], where
G is an Abelian group of order n, with cyclic codes obtained as
a special case (when G is cyclic).
• Quasi-cyclic codes: A code is t-quasi-cyclic if for every codeword (c0 ; c1 ; . . . ; cn01 ), the cyclically shifted vector by t
positions, (cn0t ; . . . ; cn01 ; c0 ; . . . ; cn0t01 ) is also a codeword
(cyclic if t = 1).
• Constacyclic codes: A code is -constacyclic if for every codeword (c0 ; c1 ; . . . ; cn01 ), the constacyclically shifted vector by
one position, ( cn01 ; c0 ; . . . ; cn02 ) is also a codeword, where
is an element of Fq3 = Fq n f0g (cyclic if = 1). These are
the ideals in the ring Fq [x]=hxn 0 i.
The motivation for studying each of these classes of generalizations
is that better codes than cyclic codes in the sense of larger minimum
distance have been found in these generalizations. The class of quasicyclic codes [1], [2] have been shown to contain asymptotically good
codes [2]. They provide link between block codes and convolutional
codes [3] and recently it has been shown that quasi-cyclic codes have
close relationship with the tail-biting representations of general block
codes [4]. Constacyclic codes over finite fields Fp (also known as
pseudocyclic code) have been extensively studied [5]–[8]. The class of
constacyclic codes includes as subclasses the important class of cyclic
codes and the class of negacyclic codes. It was shown in [5] that constacyclic maximum-distance separable (MDS) codes exist for certain
Manuscript received July 13, 2002; revised October 12, 2003. This work was
supported in part by CSIR, India, under Research Grant 22(0298)/99/EMR-II
to B. S. Rajan. The material in this correspondence was presented in part at
the IEEE International Symposium on Information Theory, Yokohama, Japan,
June/July 2003.
The authors are with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore-560012, India (e-mail:
kirant@protocol.ece.iisc.ernet.in; bsrajan@ece.iisc.ernet.in).
Communicated by C. Carlet, Associate Editor for Coding Theory.
Digital Object Identifier 10.1109/TIT.2003.822619
parameters for which cyclic MDS codes do not exist. An attraction
with constacyclic codes is that, these codes can be studied using the
well-known “theory of cyclic codes.” Also, these codes can be decoded
with slight modifications of efficient decoding algorithms for cyclic
codes. Thus, constacyclic codes can be an alternative to cyclic codes
for certain parameters.
Fig. 1 illustrates the various generalizations discussed above. Since
a finite Abelian group G is a direct product of cyclic groups, we can
view Abelian codes as multidimensional cyclic codes. This generalization of cyclic to Abelian codes can be carried over to constacyclic
codes to give multidimensional constacyclic codes. Similarly, the
generalization from cyclic to quasi-cyclic codes can be carried over
to constacyclic codes to yield quasi-twisted (QT) codes [9]–[11],
i,e., a code is ( ; t)-QT if for every codeword (c0 ; . . . ; cn01 ),
( cn0t ; cn0t+1 ; . . . ; cn01 ; c0 ; . . . ; cn0t01 ) is also a codeword.
Some QT codes with better parameters than any known codes were
listed in [9] and [10].
In this correspondence, we investigate multidimensional constacyclic codes (henceforth referred to as consta-Abelian codes) using
a twisted DFT and characterize codes that are invariant under two
additional scaled permutations (called monomials in [2]).
For a prime p, Galois rings are residue class polynomial rings
Zp [x]=(x), where Zp [x] is the ring of polynomials over Zp and
(x) is a basic irreducible polynomial of degree l over Zp [x] [12].
This Galois ring denoted by GR (pa ; l), coincides with the finite field
Fp when a = 1 and the integer residue class ring Zp when l = 1.
Recently, various aspects of coding and cryptography have been dealt
with in the general setting of Galois rings instead of finite fields. In
view of this, in this correspondence, the codes we discuss are over
Galois rings GR (pa ; l).
Permutation groups of cyclic codes over Galois rings have been investigated in [13]. In [14], we used a transform approach to study
Abelian codes over Galois rings closed under certain permutations.
The results of this correspondence are generalizations of the results
for Abelian codes discussed in [14]. Different decoding algorithms for
codes over Galois rings have been studied [15]–[17]. In certain cases,
the consta-Abelian codes belong to the class of Alternant codes and
hence the algorithm discussed in [16] could be used for decoding such
codes.
0018-9448/04$20.00 © 2004 IEEE
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
Definition 1: Let C1 and C2 be codes of length n over GR (pa ; l).
We say that C1 and C2 are equivalent if there are n permutations (0) ; (1) ; . . . ; (n01) of GR (pa ; l) and a permutation of
In = f0; 1; . . . ; n 0 1g such that, if
~c = (c0 ; c1 ; . . . ; cn01 ) 2 C1
then
(~c) = ((0) (c (0) ); (1) (c (1) ); . . . ; (n01) (c (n01) )) 2 C2 :
When all (i) ’s are identity permutations, we say that C1 and C2 are
permutation equivalent and when each (i) is a multiplication by a
nonzero scalar, C1 and C2 are said to be scalar multiple equivalent or
monomial equivalent. A code C is said to be -invariant if (~c) 2 C
for all ~c 2 C .
Monomially equivalent codes have the same weight enumerator, in
particular they have the same minimum distance.
Example 1: A -constacyclic code over GR (pa ; l) is -invariant
where is a permutation of In which takes i to (i + 1) modulo n and
is such that
(~c) = ((0) (c0 ); (1) (c1 ); . . . ; (n01) (cn01 ))
where (0) (c0 ) = c0 and () (c ) = c for = 1; . . . ; n 0 1.
To generalize the class of constacyclic codes to consta-Abelian
codes, we use the following notations. Let m0 ; m1 ; . . . ; mr01 be
nonzero positive integers and let n = m0 m1 1 1 1 mr02 mr01 . Using
m ’s, 0 r 0 1 as mixed radixes, any integer i 2 In can be
uniquely expressed as
i = i0 + i1 m0 + 1 1 1 + ir02 (m0 m1 1 1 1 mr03 )+ ir01 (m0 m1 1 1 1 mr02 )
where 0 i < m . The mixed-radix representation of i is denoted
by i = die = dir01 ; ir02 ; . . . ; i0 e and the mixed-radix addition and
subtraction, denoted by 8 and 9, respectively, are defined by
i 8 j = dar01 ; ar02 ; . . . ; a0 e
a = (i + j ) mod m ;
8
and
i 9 j = dar01 ; ar02 ; . . . ; a0 e
a = (i 0 j ) mod m ;
8 :
Let G, a finite Abelian group (with jGj = n), be a direct product of r
cyclic subgroups denoted by Cr01 ; . . . ; C0 of orders mr01 ; . . . ; m0 ,
respectively. If g(m ) ; . . . ; g(m ) are the generators of the corresponding cyclic subgroups, then any element g 2 G can be written
as
g = g(im
for some ir01 ; . . . ; i0 where
0 i < m
gi
) (m
)
1 1 1 gim
(
)
for = 0; 1; . . . ; r 0 1:
This element is denoted by gi or gdie , where
die = dir0 ; ir0 ; . . . ; i e
mixed-radix representation of i 2
1
2
used in defining the twisted-multiplication.
A -consta-Abelian code is an ideal in the twisted group ring
GR (pa ; l) [G], with the twisted-multiplication defined by =
( r01 ; r02 ; . . . ; 0 ). For every codeword ~c = (c0 ; c1 ; . . . ; cn01 )
in the -consta-Abelian code (corresponding to g 2G ci gi in the
corresponding ideal of GR (pa ; l) [G]), the element
(~c) = ((0) (c08j ); (1) (c18j ); . . . ; (n01) (c(n01)8j ))
(corresponding to twisted-multiplication of
g 2G ci gi by gj =
j
j
g(m ) g(m ) 1 1 1 g(jm ) ) also belongs to the code for all values of
j = 0; 1; . . . ; n 0 1, where
(i) (ci8j ) =
and k is such that
0
r 1
=0
i + j = m k + (i 8 j );
k
ci8j
(1)
for all = 0; 1; . . . ; r 0 1:
It is important to notice that the permutations (i) are induced because of the permutation and the twisted multiplication defined in
GR (pa ; l) [G]. In fact, (i) used in (1) can be different for different
values of j . Since we are concerned with codes which are ideals in the
twisted group ring, for any other permutation on In also, there are a set
of permutations on GR (pa ; l) uniquely associated with it. Therefore,
henceforth we denote the monomial as , and the action of this
monomial is denoted, as
(~a) = ((0) (a (0) ); . . . ; (n01) (a (n01) )):
In particular, the monomial associated with the permutation i 7! i 8 j
in (1) is denoted as 8j and the monomial for i 7! i 9 j as 9j .
In this correspondence, we are interested in the following two permutations on In .
1
Definition 2: For n0 = 1 or some ns = s0
=0 m , let Qs : In !
In , i 7! (i + ns ) modulo n and let Q denote the associated monomial. If ~a = (a0 ; a1 ; . . . ; an01 ) 2 GR (pa ; l)n , using the one to one
mapping from GR (pa ; l)n to the twisted-group ring GR (pa ; l) [G]
0
n 1
0
In using mr01 ;
mr02 ; . . . ; m0 as the mixed radixes. The group operation of G
can thus be specified using mixed-radix indexing as gi gj = gi8j ,
where i; j 2 In and i 8 j and i 9 j are the mixed-radix addition
and subtraction, respectively. Let e be the exponent of G and p be a
01 mi for all
prime such that (e; p) = 1 and, henceforth, let n = i=0
= 1; 2; . . . ; r 0 1; n0 = 1 by convention and let q = pl .
A group ring GR (pa ; l)[G] is the set of all formal sums of the
form g 2G ai gi , where ai 2 GR (pa ; l), with the usual “polynomial
is the
addition” and “polynomial multiplication.” By associating a vector
~a = (a0 ; a1 ; . . . ; an01 ) 2 GR (pa ; l)n with the element of the group
ring g 2G ai gi , an Abelian code corresponds to an ideal of a group
ring GR (pa ; l)[G]. For every codeword (c0 ; c1 ; . . . ; cn01 ) in the
Abelian code, (c08j ; c18j ; . . . ; c(n01)8j ) also belongs to the code
for all values of j = 0; 1; . . . ; n 0 1.
As in [18], we define the twisted group ring GR (pa ; l) [G]
by taking the GR (pa ; l)-module GR (pa ; l)[G] and imposing the
following twisted-multiplication on each of the generators g(m ) of
the cyclic subgroup C , given by
+j
k t
g(im ) : g(jm ) = (g(im ) ; g(jm ) )g(im
) = g(m )
where for all is an element of the unique cyclic subgroup A GR (pa ; l)3 , j A j= pl 0 1, and is a map from C 2 C to A, which
maps (g(im ) ; g(jm ) ) to k , where k is such that i + j = m k + t,
0 t < m . Let = ( r01 ; r02 ; . . . ; 0 ) be the set of all ’s
i=0
(i)
Q
a(i0n
=
0
n 1
i=0
i
ai g(m
)
i
g(m
+
)
g
i
) (m
)
1 1 1 gim
+
(
)
1 1 1 gim
(
gi
) (m )
g(im+1) 1 1 1 g(im
1 1 1 gim
(
)
)
= Q (~a)
(s)
where appears as a “carry value” added to i in the mixed-radix
representation of di + ns e
di + ns e = d(ir0
1
+ r(s0)1 )m
(is+1 + ; . . . ; (i + (s) )m ; . . . ;
)
; (is + 1)m ; is01 ; . . . ; i0 e (2)
(s)
s+1 m
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
Fig. 2.
-invariant, -invariant consta-Abelian codes.
(i)
(where (x)m denotes x modulo m), and Q (an0n
the twisted-multiplication is given by
(i)
Q
(an0n
0
r 1
+i
)=
=1
r 1
where k is such that i +
= 0; 1; . . . ; r 0 1,
g(im ) ; g(m
0
=
=1
(s)
=
k
an0n
)
+i
) induced by
an0n
0
=0
+i
m k + t , 0 t < m for all
Q (~c) = (24c6 ; 24c7 ; 24c8 ; 24c9 ; 24c10 ; 24c11 ; c0 ; c1 ; . . . ; c5 )
Q (~c) = (7c9 ; 7c10 ; 7c11 ; c0 ; c1 ; c2 ; 18c3 ; 18c4 ; 18c5 ; c6 ; c7 ; c8 )
Q (~c) = (18c11 ; c0 ; c1 ; 24c2 ; c3 ; c4 ; 7c5 ; c6 ; c7 ; 24c8 ; c9 ; c10 )
12
where ~c = (c0 ; c1 ; c2 ; . . . ; c10 ; c11 ) 2 Z25
.
Definition 3: Let b = dbr01 ; br02 ; . . . ; b0 e be such that
gcd(b ; m ) = 1 for all = 0; 1; . . . ; r 01. Let Ub : In ! In be a map,
defined by
die = dir0 ; ir0 ; . . . ; i e
!d(br0 ir0 )m
2
0
1
1
in the integer ring modulo the respective mixed radix.). The associated
(i)
monomial is denoted as U , and the action of U is scalar multiplication by
r 1
+i
Example 2: For n = 22223 length codes over Z25 = f0; 1; . . . ; 24g
and = (24; 18; 24), the monomials defined above for different values
of s are given by
1
369
; (br02 ir02 )m
; . . . ; (b0 i0 )m
e
where (x)m denotes x modulo m. We call this permutation the Unitb
permutation. (Notice that every mixed-radix component of b is a unit
k
where k satisfies b i = m k + t ;
0 t < m :
Example 3: For n = 22223 length codes over Z25 , = (24; 18; 24)
and b = d1; 1; 5e, the monomial defined above is given by
U (~c) = (c0 ; 24c2 ; 24c1 ; c3 ; 24c5 ;
24c4 ; c6 ; 24c8 ; 24c7 ; c9 ; 24c11 ; 24c10 )
12
for ~c = (c0 ; c1 ; c2 ; . . . ; c10 ; c11 ) 2 Z25
.
In this correspondence, we study -consta-Abelian codes
which are also invariant under the monomials Q and U .
These codes are called Q -invariant -consta-Abelian codes and
U -invariant -consta-Abelian codes, respectively. The classes
of codes studied in this correspondence are best explained with
Fig. 2, where, the class of -consta-Abelian codes of length n =
mr01 mr02 1 1 1 m1 m0 is depicted by the ellipse in the figure. We
show that, in the class of linear codes over GR (pa ; l) of length
n = mr01 mr02 1 1 1 m1 m0 , a Q -invariant code is also Q -invariant for every s r 0 1. We also show that every
-consta-Abelian code of length n = mr01 mr02 1 1 1 m1 m0 is
( r01 ; nr01 )-QT. Hence, the ellipse has not gone outside the
concentric circles. The circle (shown in bold) represents n-length
U -invariant linear codes (not necessarily -consta-Abelian). The
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
horizontally hatched regions represent Q -invariant -consta-Abelian
codes for some value of s and double-hatched regions represent
U -invariant and Q -invariant -consta-Abelian codes.
The main result of this correspondence consists of: i) characterizing
Q -invariant and U -invariant -consta-Abelian codes over Galois
rings and ii) finding the smallest value of such s and all values of
b for every -consta-Abelian code. We obtain these results using the
DFT approach. DFT domain characterization of cyclic, Abelian, and
quasi-cyclic codes over finite fields and rings Zm have been previously
discussed in the literature [19]–[22] and a similar technique has been
used to characterize constacyclic codes in [23]. In this correspondence,
we generalize the results of [23] for consta-Abelian codes over Galois
rings and, then, characterize Q -invariant -consta-Abelian codes and
U -invariant -consta-Abelian codes over Galois rings in the DFT domain defined over suitable Galois ring extensions. As a consequence,
we characterize -consta-Abelian codes that are both Q -invariant
and U -invariant. Duals of these codes are shown to be invariant under
both monomials Q and U .
If C is an arbitrary ( r01 ; nr01 )-QT code of length n =
mr01 mr02 1 1 1 m0 over GR (pa ; l), then the extension ring required for transform domain description of this code is GR (pa ; lm)
which should contain an nth root of unity as well as an nth root
of r01 (in other words, m is the smallest integer such that both
n and order ( r01 ) 2 n divide plm 0 1). But, if this code C is
-consta-Abelian also (see Fig. 2), then an extension ring containing
m th roots of unity and m th roots of for all = 0; 1; . . . ; r 0 1 is
enough, which is generally smaller than GR (pa ; lm). Since algebraic
decoding generally takes place in the extension ring, a smaller extension ring may lead to a simpler or more efficient decoding of code C .
Throughout this correspondence, the length of the code is relatively
prime to p, where pa is the characteristic of the Galois ring GR (pa ; l)
over which the code is defined.
The remainder of this correspondence is organized as follows. In
Section II, we define a twisted DFT and characterize consta-Abelian
codes using this transform. Consta-Abelian codes invariant under Q
and U are then characterized in Sections III and IV, respectively. Examples are given to illustrate each class of code. In Section V, we discuss duals of consta-Abelian codes and obtain nonexistence result for
certain self-dual codes. Finally, the conclusions of this correspondence
are presented in Section VI.
II. TWISTED-DFT CHARACTERIZATION OF CONSTA-ABELIAN CODES
=
For a -consta-Abelian code over GR (pa ; l), ( r01 ; r02 ; . . . ; 0 ) is an r-tuple where each is an element of
the cyclic subgroup of order pl 0 1 in GR (pa ; l). Let d be the order
of element and let m be the smallest positive integer such that
m d j (qm 0 1) for all = 0; 1; . . . ; r 0 1. Let be a primitive
root of unity of order m and be an m th root of , for all ,
belonging to the extension ring GR (pa ; lm). Let GR (pa ; lm)[G] be a
group ring with “componentwise addition” (polynomial addition), i.e.,
g 2G
ai gi +
g 2G
bi gi =
(ai + bi )gi
g 2G
and “componentwise multiplication”
(
ai gi ):(
bi gi ) =
(ai bi )gi
g 2G
g 2G
g 2G
and let GR; (pa ; l) [G] be the twisted group ring defined earlier
(polynomial addition and polynomial multiplication). We now define a
group ring homomorphism from GR (pa ; l) [G] into GR (pa ; lm)[G],
and use this to characterize consta-Abelian codes over GR; (pa ; l).
A. Twisted DFT (TDFT)
Definition 4: Let ~a = (a0 ; a1 ; . . . ; an01 ) 2 GR (pa ; l)n . Let and for = 0; 1; . . . ; r 0 1, be as defined earlier. The TDFT vector
A~ = (A0 ; A1 ; . . . ; An01 ) 2 GR (pa ; lm)n of ~a is defined as
n01 r01
i i j
ai ;
Aj =
for all j 2 In
(3)
i=0 =0
where i = dir01 ; ir02 ; . . . ; i0 e and j = djr01 ; jr02 ; . . . ; j0 e are
mixed-radix representations of i and j . Let be a primitive m d th
root of unity in the cyclic group of order plm 0 1 in GR (pa ; lm) and
h
d
= , = for all = 0; 1; . . . ; r 0 1. Then the above
TDFT can be expressed as
n01 r01
Aj =
i (h +d j ) ai ;
for all j 2 In :
(4)
i=0 =0
Henceforth, we refer to any ~a 2 GR (pa ; l)n as “time domain
~ 2 GR (pa ; lm)n as TDFT vector
vector” and the corresponding A
of ~a. Notice that, if = 1 for all , then the ideals correspond to
n = mr01 mr01 1 1 1 m0 length Abelian codes over GR (pa ; l) and
the TDFT reduces to the generalized DFT defined in [14]. The TDFT
satisfies the following properties:
n01 r01
r01
0i
0i j Aj
ai = n01
(5)
j
=0
=0
=0
n01
ci =
9(ik) (ai9k ) bk tdft
!Aj Bj
(6)
k=0
where ! denotes a TDFT pair. In (6), the left-hand side is the
twisted convolution, which corresponds to the “polynomial multiplication” of two elements in GR (pa ; l) [G]. This maps to the componentwise multiplication of the corresponding TDFT vectors in the group
ring GR (pa ; lm)[G].
Let 0 be the Frobenius automorphism of GR (pa ; lm) [12]. Using
p-adic representation, an element 2 GR (pa ; lm) can be represented
as 0 + 1 p + 1 1 1 + a01 pa01 where i are elements from the Teichmuller set T = GR (pa ; lm)=p GR (pa ; lm) (isomorphic to the field
Fp ). Then 0 ( ) is equal to 0p + 1p p + 1 1 1 + ap01 pa01 . Alternately,
if polynomial representation is used for elements of Galois ring, then
each element of the set T is a polynomial in x. Therefore, 0 can be
uniquely defined by defining its action on x. Let = 0l , then is an
automorphism of GR (pa ; lm) that fixes GR (pa ; l) GR (pa ; lm).
tdft
Lemma 1: (Conjugate symmetry property) If
A~ = (A0 ; A1 ; . . . ; An01 ) 2 GR (pa ; lm)n
is the TDFT vector of (a0 ; a1 ; . . . ; an01 ) 2 GR (pa ; l)n , then the following relation among Aj ; j 2 In , holds:
k (Aj ) = Ai
(7)
where i = dir01 ; ir02 ; . . . ; i0 e with
k
i = (q 0 1)h + qk j modulo m
d
for all = 0; 1; . . . ; r 0 1.
Let k denote a mapping from Im to Im which maps j to
(qk 0 1)h + qk j :
d
Let 8k denote the mapping from In to In , which maps i =
dir01 ; ir02 ; . . . ; i0 e to
8k (i) = dkr01 (ir01 ); rk02 (ir02 ); . . . ; 0k (i0 )e:
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
TABLE I
TDFT PARAMETERS FOR n = 2 2
2 2 2 AND = (1; 1; 4)
Definition 5: For every j
2 In , the set
j; 8 (j ); 82 (j ); . . . ; 8e 01 (j )
where ej is the smallest integer such that 8e (j ) = j , is called the
cyclotomic coset modulo n containing dj e and ej is called the exponent of dj e. Clearly, In is a disjoint union of such cyclotomic cosets.
Let L In be the set containing one element from each of the cyclotomic cosets. We call the set L the cyclotomic representative set. (For
dj e =
1
concreteness, we use the smallest element of a cyclotomic coset as a
representative.)
Example 4:
i) Table I lists the TDFT parameters for n = 2 2 2 2 2 length
(1; 1; 4)-consta-Abelian codes over F5 . The cyclotomic cosets each
consist of a single element and hence the corresponding representative set L = I8 .
ii) For n = 2 2 2 2 3 length (24; 18; 24)-consta-Abelian codes over
Z25 , the smallest extension required for the TDFT description is the
Galois ring
GR (52 ; 2) = Z25 [x]=(x2 + x + 2) = fc + dx j c; d 2 Z25 g:
The Frobenius automorphism 0 is given by 0 (x) = 24 + 24x. Z25
is the only nontrivial Galois subring of GR (52 ; 2). It can be checked
that all the elements of Z25 are invariant under the Frobenius automorphism. The only nontrivial ideal of Z25 is 5Z25 which is equal to
f0; 5; 10; 15; 20g and the only nontrivial ideal of GR (52 ; 2) is
5 GR (52 ; 2) = fc + dx j c; d 2 5Z25 g:
The group of units
= GR (52 ; 2) n 5GR(52 ; 2):
The cyclic subgroup of order plm 0 1 = 24, G1 GR (52 ; 2)3 with
GR (52 ; 2)3
elements listed in the increasing order of powers of generator element
6 + x is
G1 = f6 + x; 9 + 11x; 7 + 14x; 14 + 2x; 5 + 24x;
7; 17 + 7x; 13 + 2x; 24 + 23x; 23 + 14x;
10 + 18x; 24; 19 + 24x; 16 + 14x;
18 + 11x; 11 + 23x; 20 + x; 18; 8 + 18x;
12 + 23x; 1 + 2x; 2 + 11x; 15 + 7x; 1g:
Table II(a) lists the TDFT parameters, Table II(b) lists the cyclotomic
cosets, and Table II(c) is the corresponding representative set L .
The constraint due to the conjugate symmetry property given by (7)
implies that: i) the set of transform components fAdj e j dj e 2 dieg
denoted by Adie are related (in other words, transform components indexed by elements of the same cyclotomic coset are related) and moreover ii) every element of Adie belongs to the same Galois ring
GR (pa ; lei ) GR; (pa ; lm)
for some fixed ei dividing m. The set Adie will be called the conjugacy
class containing Adie . For a code C over GR (pa ; l), let C j = fAj j
8~a 2 Cg denote the set of distinct values taken by the j th transform
component of all the codewords in C and let
C i;j = f(Ai ; Aj ) j 8~a 2 Cg:
Then, a -consta-Abelian code C over GR (pa ; l) is isomorphic to an
ideal i2L C i , where C i = p GR (pa ; lei ) for some fixed value
of i , 0 i a and transform components belonging to different
371
conjugacy classes take values independently. Here, Ai and Aj take
values independently implies C i;j = C i 2 C j .
In the remainder of this correspondence, we refer to the ideal
p GR (pa ; lei ) as the i -ideal of GR (pa ; lei ). Also, for a -constaAbelian code, since Adie can take values only from the ideals of Galois
subring GR (pa ; lei ), we will say Adie takes values from the i -ideal
to mean that C i = p GR (pa ; lei ), since it is obvious which ring is
meant. Hence, a -consta-Abelian code is specified/characterized by
specifying i ; i ; . . . ; i corresponding to each element in L =
fdi1 e; di2 e; . . . ; dijLj eg. In other words, a -consta-Abelian code
over GR (pa ; l) can be characterized by a partition of In as given in
the following Definition 6.
Definition 6: The defining partition of a -consta-Abelian code is
the partition (T0 ; T1 ; . . . ; Ta ) of In , where
T = fj 2 In j C j = p GR (pa; lej )g;
for 0 a:
For a -consta-Abelian code, every T is a union of some cyclotomic
cosets or T = the empty set ; if C j 6= p GR (pa ; lej ) for any j 2 In .
This partition uniquely specifies the -consta-Abelian code.
Example 5:
i) Some (1; 1; 4)-consta-Abelian codes of length n = 2 2 2 2 2
over F5 are listed in Table III. In the TDFT description of all
codes, only the transform components Aj for which C j 6= 0
are listed. The corresponding TDFT parameters and cyclotomic
cosets are given in the previous example.
ii) Table IV lists some n = 2 2 2 2 3 length (24; 18; 24)-constaAbelian codes over Z25 . The corresponding TDFT parameters
and cyclotomic cosets are given in the previous example.
iii) (24; 18; 24)-consta-Abelian codes over GR (52 ; 2) of length
n = 2 2 2 2 3 are listed in Table V. In these tables, an element
a + bx of the Galois ring GR (52 ; 2) is listed as a 2-tuple
adjacent to each other. For this case, the extension ring required
is the Galois ring GR (52 ; 2) (see Example 4, part ii)) over
which the code is defined and hence all cyclotomic cosets
contain single elements.
B. Constraints on L
The main result of this correspondence is to identify the constraints
on the values taken by transform components belonging to different
conjugacy classes for the -consta-Abelian code to be
1) U -invariant for some particular values of
b = dbr01 ; br02 ; . . . ; b0 e
such that gcd(b ; m ) = 1 for all = 0; 1; . . . ; r 0 1 and
2) Q -invariant for any s, 0 s r 0 1.
Given the transform domain description of a -consta-Abelian code,
this result enables us to give the smallest value of s for which the code is
Q -invariant and also all the values of b for which the code is U -invariant. Toward this end, we define a constraint in terms of a partition
on L and -consta-Abelian codes satisfying this constraint as follows.
Definition 7: A constraint D = fD1 ; D2 ; . . . ; Du g is a partition of
the set of cyclotomic coset representatives
L = fdi1 e; di2 e; . . . ; dijLj eg:
A -consta-Abelian code over GR (pa ; l) is said to satisfy the constraint D if dic e; did e 2 Dj for some j 2 f1; 2; . . . ; ug implies i =
i , where C i = p GR (pa ; lei ) and C i = p GR (pa ; lei ). If
a set Dj contains only one cyclotomic coset representative, we call the
corresponding cyclotomic coset a free cyclotomic coset. Otherwise,
Dj is called a constrained set of cyclotomic coset representatives
and all the corresponding cyclotomic cosets of Dj are said to form a
constrained set.
372
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
TABLE II
TDFT PARAMETERS FOR n = 2 2 3 AND = (24; 18; 24)
2 2
(a) TDFT parameters.
(c) Cyclotomic coset representative set L .
(b) Cyclotomic cosets.
TABLE III
n = 2 2 2 2 2 LENGTH (1; 1; 4)-CONSTA-ABELIAN CODES OVER F
From Definition 6, if (T0 ; T1 ; . . . ; Ta ) is the defining partition
of a -consta-Abelian code C , then, C satisfies the constraint
D = fD1; D2 ; . . . ; Du g if and only if every set T is either empty or
a union of some constrained sets Dj .
where s~a = ( s a0 ; s a1 ; . . . ; s an01 ). Since the code
GR (pa ; l)-linear, s01 (Q )m (~a) 2 C .
Remark 1: We had earlier introduced the notion of transform components belonging to different conjugacy classes taking values independently for a -consta-Abelian code. The above definition implies
that if a -consta-Abelian code satisfies constraint D , and if i1 and i2
from two different cyclotomic cosets belong to the same set Dj , then
i = i = j but C i ;i = C i 2 C i continues to be true. If i1
and i2 belong to different sets in D , then i ; i can be different and
C i ;i = C i 2 C i .
discussed in [14].
For the ideals in the twisted group ring GR (pa ; l) [G], corresponding to the permutation Qs , we have a monomial Q induced
by the twisted-multiplication defined earlier. In this section, we
characterize all -consta-Abelian codes which are invariant under the
monomial Q .
Proposition 1: A Q -invariant linear code C (over GR (pa ; l)) is
Q -invariant as well.
Proof: Let (Q )t denote the composition of Q with itself, t
times. Since C is Q -invariant, (Q )t (~a) 2 C for any ~a 2 C . For
t = ms
(Q )m (~a) = Q
( s~a) = s (Q
(~a)) 2 C
is
For the special case when k = 1, for all 0 k r 0 1, the
Q -invariant -consta-Abelian codes are exactly the ns -QCA codes
Theorem 1: All n = mr01 mr02 1 1 1 m0 length -consta-Abelian
-invariant.
codes are Q
Proof: Let C be a -consta-Abelian code over GR (pa ; l). For any
~a =
the codeword
g(m
III. Q -INVARIANT CODES
C
n01
)
i=0
ai gi =
n01
i=0
n01
i=0
ai gi
(i)
Q
2C
(ai8dm
01;0;0;...;0e )gi
= Q (~a) 2 C :
Notice that the action of the monomial Q
is
Q (~a) = ( r01 an0n ; . . . ; r01 an01 ; a0 ; a1 ; . . . ; an0n 01)
which is essentially r01 -constacyclic shifts by nr01 positions.
Hence, all -consta-Abelian codes are ( r01 ; nr01 )-QT as indicated
in Fig. 2.
In the next few theorems, we will use the following notations.
For = ( r01 ; r02 ; . . . ; 0 ), let be the smallest integer
(0 < r) such that r01 = r02 = 1 1 1 = +1 = 1 and 6= 1.
By convention, = 01, if = (1; 1; . . . ; 1).
Definition 8: For every j 2 In such that
dj e = d0; 0; . . . ; 0; j 6= 0; j01 ; . . . ; j0 e
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
373
TABLE IV
n = 2 2 2 2 3 LENGTH -CONSTA-ABELIAN CODES OVER Z
(i.e., j is the first nonzero mixed-radix component) and > s (h;s)
(j ) be as defined in (8) at the top of the
0; h > s 0, let the set J
following page. Notice that h can be greater than, less than, or equal
to . If h = s + 1, we denote the above set as J (s) (j ) for notational
simplicity, which is the set of all die 2 In with only the sth component
running over Im and for all 6= s, i = j .
Definition 9: Let
L = fdi1 e; di2 e; . . . ; dijLj eg:
For any s, 0 s < r 0 1, and any , 01 r 0 1, and die 2 L
such that die = d0; . . . ; 0; i 6= 0; i01 ; . . . ; i0 e, the subset diehsi of
L is defined as in (9) at the top of the following page. Since for any
pair of dic e; did e 2 L , dic ehsi and did ehsi are either same or disjoint,
fdiehsi jdie 2 Lg constitute a partition of L. This partition of L will
be called the s-partition of L .
FOR
= (24; 18; 24)
Example 6: i) Table VI lists the s-partitions for 2 2 2 2 2 length
codes over F5 . ii) The cyclotomic cosets and
various constrained sets for n = 2 2 2 2 3 length codes over Z25
are given in Table VII(a) and (c). iii) Table VIII(a) and (b) lists the
cyclotomic cosets and constrained sets for n = 2 2 2 2 3 length codes
over GR (52 ; 2).
(1; 1; 4)-consta-Abelian
Lemma 2: If C is an n = mr01 mr02 1 1 1 m0 -length -consta= p GR (pa ; lek ) for each
Abelian code such that C k
k = djr01 ; jr02 ; . . . ; ks ; . . . ; j0 e 2 J (s) (j ), then
k2J
k
s s
Ak 2 p
~ 2 TDFT (C )
GR (pa; le) for every A
(j )
if and only if k and ek je for all k 2 J (s) (j ).
Proof: Similar to the proof of Lemma 1 in [14].
374
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
J (h;s) (j ) = fd0; . . . ; 0; j ; j01 ; . . . ; jh ; xh01 ; xh02 ; . . . ; xs ; js01 ; . . . ; j0 ejx 2 Im ; = h 0 1; h 0 2; . . . ; sg:
fig;
if s ; s.
(;s)
j
2
L
j
d
j
e
2
J
d
i
e
;
for
some
if > s or > s
an
element
of
(
k
)
k
2
diehsi =
j 2 L j an element of dj e 2 J (;s) (k) for some k 2 die ; if < ; s < n01 r01
i
i j
A(js) =
Q(i) (ai0n )
=
i=0
n 1
0
=0
r 1
0
(
j
i=0 =s+1
n 1n 1
r 1
= n1
= n1
0 0
k=0 i=0
2
k J
0
(
=s+1
n 1
0
(j )
i=0
)(i
+
j
(
)
)
j
s s
)(i +1)
j
s s
j
0
1
=s+1
0
r 1
and p such that gcd(n; p) = 1, a length
n = mr01 mr02 1 1 1 m0 -consta-Abelian code over GR (pa ; l) is
Q -invariant for = (1; . . . ; 1; 6= 1; . . . ; 0 ) and 0 s r 0 2,
if and only if it satisfies the constraint D = fD1 ; D2 ; . . . ; Du g, where
j 2 Di L ) Di = dj ehsi .
Proof: Let C be a -consta-Abelian code and let ~a 2 C . For C to
~ (s) = (A(0s) ; A(1s) ; . . . ; A(ns0) 1 )
be Q -invariant, Q (~a) 2 C . Let A
be the TDFT-vector of Q (~a).
Let d0; 0; . . . ; 0; j ; . . . ; j0 e 2 L . In the next set of equations, we
assume that > ; > s. For the other cases, the procedure remains
(s)
n
same. Using the TDFT, we can write Aj as in (10) at the top of this
page and then inverse TDFT is used to reduce (11) to (12), also at the
top of the page.
For the particular case s = r 0 2, continuing from (13) (at the top
of the page), the complete proof for this Theorem is given in the Appendix. For 0 s < r 0 2, (13) can be further reduced to (14) (see
(s)
(14) and (15) at the top of the following page). In (14), since s+2 = 0,
(s)
if s+1 = 0 or if is+1 6 01 mod ms+1 , we can further split the first
part of the right-hand side of (14) and obtain (15). Observe that K is
independent of js and K1 is independent of both js and js+1 .
Having proved Theorem 2 for s = r 0 2 separately (see the Appendix), we can use this as the induction base together with Proposition
1, and the remaining part of the proof is similar to the corresponding
proof of Theorem 3 in [14].
Example 7:
i) Table VI lists the constrained sets for (1; 1; 4)-consta-Abelian
codes. The monomial Q acting on a codeword for different
values of s are:
Q (c0 ; c1 ; . . . ; cn01 ) = (c4 ; c5 ; c6 ; c7 ; c0 ; c1 ; c2 ; c3 ) :
Q (c0 ; c1 ; . . . ; cn01 ) = (c6 ; c7 ; c0 ; c1 ; c2 ; c3 ; c4 ; c5 ) :
Q (c0 ; c1 ; . . . ; cn01 ) = (4c7 ; c0 ; 4c1 ; c2 ; 4c3 ; c4 ; 4c5 ; c6 ) :
In Table III, codes C0 , C1 , and C8 are Q -invariant; codes C2 ,
C3 , and C9 are Q -invariant; and rest of the codes are Q -in-
variant.
ii) Table IV lists some (24; 18; 24)-consta-Abelian codes over Z25
for the constrained sets listed in Table VII(c). The monomial
Q for different values of s are
Q (~c)
=0
i (j
0k
)
i (j
0k
)+
(9)
(10)
ai
i j
i
=0
Using Lemma 2, we will next prove one of the main results (Theorem
2) of this correspondence. The outline of the proof is similar to [14,
proof for Theorem 3 ] (we use induction on s) and, therefore, we give
complete proof only for s = r 0 2 in the Appendix and for Theorem
2, we give only a part of the proof that highlights the differences.
Theorem 2: For any
0
s 1
(8)
(11)
Ak
(12)
j
i (j
s s
0k
)+j
Ak :
(13)
= (24c6 ; 24c7 ; 24c8 ; 24c9 ; 24c10 ; 24c11 ; c0 ; c1 ; . . . ; c5 ) :
Q (~c)
= (7c9 ; 7c10 ; 7c11 ; c0 ; c1 ; c2 ; 18c3 ; 18c4 ; 18c5 ; c6 ; c7 ; c8 ) :
Q (~c)
= (18c11 ; c0 ; c1 ; 24c2 ; c3 ; c4 ; 7c5 ; c6 ; c7 ; 24c8 ; c9 ; c10 ) :
Among codes in Table IV, C1 and C4 are Q -invariant and the
remaining codes are Q -invariant.
iii) Table V lists some Q -invariant (24; 18; 24)-consta-Abelian
codes over GR (52 ; 2) for constrained sets in Table VIII(b). The
monomials Q for different values of s are the same as those
for codes over Z25 given above.
U -INVARIANT CODES
Let b = dbr01 ; br02 ; . . . ; b0 e be such that gcd(b ; m ) = 1 for all
= 0; 1; . . . ; r 0 1. Let Ub : In ! In , which takes
die = dir01 ; ir02; . . . ; i0e ! dbr01 ir01; br02 ir02; . . . ; b0 i0e:
1
01
01
Let dbe01 = db0
r01 ; . . . ; b0 e, where b represents the inverse of b
in Im . In this section, we characterize U -invariant -consta-Abelian
codes for a special case of b defined above, which also satisfies: d j
(b01k;b
0 1) for all = 0; 1; . . . ; r 0 1. For every = 0; 1; . . . ; r 0 1,
let denote a mapping from Im to Im which maps j to
(b0k 0 1)h + b0k j
d
and let 8k;b be a map which maps dj e to
k;b
k;b
(j0 )e:
dk;b
r01 (jr01 ); r02 (jr02 ); . . . ; 0
k;b
k
Notice that for dbe = dq; q; . . . ; q e the maps = and 8k;b =8k ,
IV.
which is used to define the cyclotomic cosets.
2 In and b as defined above, let
die
= fdie; 8 ;b (die) ; 8 ;b (die) ; . . . ; 8e 0 ;b (die)g
where e0i is the smallest integer such that 8e ;b (die) = die.
Definition 10: For any i
(b
)
1
2
1
(16)
Definition 11: Let
For every die
defined to be
2
L = fdi1 e; di2 e; . . . ; dijLj eg:
L, the associated subset of L , denoted by die(b) , is
die b = dj e 2 Ljdj e = dke for somek 2 die b
(
( )
)
:
(17)
Note that with the definition above, clearly, every b defines a partition
on L .
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
Aj(s) = 1
n
+
A(js) =
0
n 1
k2J
i=0;
(j )
0
n 1
2
k J
1
+1
+
=
ms
0 ms m1 s
j
s s
2
k J
2
k J
0k
Ak
)
=s+2
i (j
0k
)+
j
j
Ak :
0k
)+
j
j
Ak :
(14)
k
s s
Ak
(j )
Ak
(j )
k
s s
i=0;
(j )
0
1
k
s+1 s+1
k
s s
0
2
0 1)
j
s+1 s+1
n 1
k J
i (j
=1
(
1
n
=s
k
s+1 s+1
k
s s
i=0;
(j )
Aj +
j
s s
0
1
j
s+1
=0
1
n
s+1
j
s s
375
0
1
k
s+1 s+1
=s+2
i (j
=1
Aj + K + K1
(15)
where
K= 1 (
ms
0 1)
j
s+1 s+1
n
2
k J
0
n 1
1
K1 =
k
s s
2
k J
(j )
0 ms m1 s
+1
k
s+1 s+1
k
s s
i=0;
Ak and
(j )
Aj =
(b)
k
s s
2
k J
k
s+1 s+1
=
=
=
0
0
r 1
i=0
n 1
0
=0
r 1
i=0
n 1
0
=0
r 1
i=0
n 1
0
=0
r 1
i=0
=0
= Adx
0
0
0
;x
i
i j
U (aU
i
i j
adbedie
(b
(i)
i ) (b
i (b
;...;x
h +b
i )j
d j )
(i)
)
adie
adie
e
where
x =
(b01 0 1)h
d
i (j
0k
)+
j
j
Ak
Ak
(j )
n
n 1
=s+2
=1
and p such that gcd(n; p) = 1, an n =
mr01 mr02 1 1 1 m0 length -consta-Abelian code is U -invariant if
and only if it satisfies the constraint D = fD1 ; D2 ; . . . ; Du g, where
j 2 Di L ) Di = dj e(b) .
(b)
Proof: Let Aj denote the j th component of TDFT (U (~a)).
From the TDFT expression
Theorem 3: For
0
1
+ b01 j
for all = 0; 1; . . . ; r 0 1. This implies that a -consta-Abelian code
is U -invariant if and only if C dj e = C dxe .
Corollary 1: All -consta-Abelian codes over GR (pa ; l) are
U -invariant for b = dq01 ; q01 ; . . . ; q01 e.
Example 8: Table VII(c) lists constrained sets for U -invariance
of (24; 18; 24)-consta-Abelian codes over Z25 . For b = d1; 1; 5e, the
action of monomial U for the above parameters is
U (~c)
= (c0 ; 24c2 ; 24c1 ; c3 ; 24c5 ; 24c4 ; c6 ; 24c8 ; 24c7 ; c9 ; 24c11 ; 24c10 ):
. Similarly, for b = d1; 5; 1e, the corresponding vector is
U (~c)= (c0 ; c1 ; c2 ; 24c3 ; 24c4 ; 24c5 ; c6 ; c7 ; c8 ; 24c9 ; 24c10 ; 24c11 ):
In Table IV, all codes are U -invariant for
b = d5; 1; 1e; d1; 5; 5e; d5; 5; 5e
where as codes C1 and C4 are U -invariant for
b = d1; 1; 5e; d1; 5; 1e; d5; 1; 5e; d5; 5; 1e
as well.
For = (1; 1; . . . ; 1), the results of this section match the characterization of Ub -invariant Abelian codes discussed in [14]. Notice that
Ub -invariant Abelian codes for b = dbr01 ; br02 ; . . . b0 e are also invariant under the permutation : die 7! dke, where i 7! k =
(b i + j )m for all j 2 Im , and 0 r 0 1 (i.e., an affine
permutation in each mixed-radix component). Similarly, it is not difficult to see that the U -invariant consta-Abelian codes discussed in
this section are also invariant under monomials , where is any permutation die 7! dke defined earlier.
376
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
TABLE V
n = 2 2 2 2 3 LENGTH -CONSTA-ABELIAN CODES OVER GR (5 ; 2) FOR = (24; 18; 24)
V. DUAL CODES
If C is a GR (pa ; l)-linear code, its dual
normal inner product
C? =
~y 2 GR(pa; l)n :
n01
i=0
0h 0h
C ? is defined using the
xi yi = 0; 8 ~x 2 C :
In this section, we show that the dual of a -consta-Abelian code
over GR (pa ; l) is a 01 -consta-Abelian code over GR (pa ; l) where
01 = ( r0011 ; r0012 ; . . . ; 001 ). Given a transform domain description of a -consta-Abelian code we are able to give the corresponding
description for the dual code which is a 01 -consta-Abelian code. If
h
h
= is an m th root of that is used in the -TDFT, let 0
1
0
1
-TDFT. Notice that d divides
be an m th root of used in the
h? + h for all = 0; 1; . . . ; r 0 1. Further, for a given cyclotomic
coset die, the set, which is a collection of j ? = dkr01 ; kr02 ; . . . ; k0 e
such that k =
0 j for each djr01 ; jr02 ; . . . ; j0 e 2 die, is
d
a valid cyclotomic coset for the 01 -TDFT. This is because, if die =
dir01 ; ir02 ; . . . ; i0 e
0h? 0 h 0 k (i ) = 0h? 0 h 0 (qk 0 1)h 0 qk i
d
d
=
(q k 0 1)h?
=
k
d
+q
d
0h? 0 h
d
k
0h? 0 h 0 i
0 i
d
in 01 -TDFT:
We call this cyclotomic coset as the dual cyclotomic coset of die.
Theorem 4: If (T0 ; T1 ; . . . ; Ta ) is the defining partition of a
-consta-Abelian code C over GR (pa ; l) and if die T , then its
dual code is a 01 -consta-Abelian code over GR (pa ; l) with defining
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
TABLE VI
CONSTRAINED SETS i
FOR 2
2
2 LENGTH
(1; 1; 4)-CONSTA-ABELIAN CODES
de
2 2
377
Using the TDFT expression
Cj =
=
=
=
TABLE VII
CYCLOTOMIC COSETS AND CONSTRAINED SETS
LENGTH CODES OVER Z
FOR
n
= 2
=
2223
=
where k
(a) Cyclotomic cosets.
(b) Cyclotomic coset repressentative set L .
(c) Constrained sets
de
(d) Constrained sets
i
de
i
i=0
r01
i=0
n01
=0
r01
i=0
n01
=0
r01
i=0
n01
=0
r01
i=0
n01
=0
r01
i=0
n01
=0
r01
i=0
=0
= 0h d 0h 0 j
01
i
i j
ci
i
i j
9(0)i (b0i )
0i 0i
01
for different values of s.
for different values of b.
9(ji) (bj 9i ) ai = 0;
for all j
2 In :
i=0
9(0)i (b0i ) ai = 0:
Let ~c = (c0 ; c1 ; . . . ; cn01 ) be such that ci
n01
i=0
bi
01
i
i (0h 0d
i
(h
+d
j
k
i j
bi
(18)
bi = Bk
(19)
bi
)
)
modulo m for all = 0; 1; . . . ; r 0 1.
1
In (18), is an m th root of 01 and 0
is a primitive m th
root of unity for all = 0; 1; . . . ; r 0 1. This proves that the
~ is a 01 -TDFT vector. Let C ? be the collection of all
vector C
~c = (c0 ; c1 ; . . . ; cn01 ) such that ci = 9(0)i (b0i ) for all ~b 2 C 3 .
Equation (19) proves that the defining set (T0? ; T1? ; . . . ; Ta? ) of
C ? is such that the dual cyclotomic coset of dj e belongs to Ta?0 if
dj e T in C . Further, the type (see [24]) of C ? and C 3 are the same,
with j C jj C ? j=j GR (pa ; l)n j. Hence, C ? is the dual of C .
We next show that the dual of a U -invariant (resp., Q -invariant)
-invariant (resp., Q -invariant). Before
proving this, we would like to point out that there is an abuse of notation
by using permutation U (resp., Q ) for both -consta-Abelian and
its dual, which is actually a 01 -consta-Abelian code. By definition,
both U and Q depend on the actual value of = ( r01 ; . . . ; 0 )
and, hence, strictly speaking we should have used different notations
for the monomial U (resp., Q ) acting on the -consta-Abelian code
and its dual. We avoid this by saying that the dual of a U -invariant
(resp., Q -invariant) -consta-Abelian code is a U -invariant (resp.,
Q -invariant) 01 -consta-Abelian code.
Let D = fD1 ; D2 ; . . . ; Du g be a constraint for a U -invariant
-consta-Abelian code defined in Definition 11. For any constrained
set Dj , the set Dj? , which is a collection of representative elements of the cyclotomic coset containing k? , for all k 2 Dj , is a
constrained set for a U -invariant 01 -consta-Abelian code. This
is because, from Definition 11, an element of Dj is a representative element of the cyclotomic coset containing an element of the
t;b
(i0 )e for some i 2 L,
form 8t;b (i) = dr01 (ir01 ); . . . ; t;b
0
0 t < ei , and since
0h? 0 h 0 t;b (i ) = 0h? 0 h 0 (bt 0 1)h 0 bt i
d
d
d
t
?
?
= (b 0 1)h + bt 0h 0 h 0 i
d
In particular
n01
j
-consta-Abelian code is U
partition (T0? ; T1? ; . . . ; Ta? ) such that the dual cyclotomic coset of
die is a subset of Ta?0 .
Proof: If C is a -consta-Abelian code over GR (pa ; l) with j 2
T , let C 3 be a -consta-Abelian code over GR (pa ; l) with defining set
(T03 ; T13 ; . . . ; Ta3 ) such that j 2 Ta30 . If ~a 2 C and ~b 2 C 3 , clearly,
Aj Bj = 0 for all j 2 In . By the twisted-convolution property (6), this
implies
n01
n01
ai ci = 0:
= 9(0)i (b0i ). Therefore,
=
t;b
0h? 0 h 0 i
d
d
is in 01 -TDFT, the above set Dj? is actually equal to the collection
of representative elements of 01 -TDFT cyclotomic coset containing
8t;b (i? ), 0 t < ei . This set, according to Definition 11, is a valid
constrained set for 01 -consta-Abelian code.
378
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
TABLE VIII
CYCLOTOMIC COSETS AND CONSTRAINED SETS FOR n = 2
2 2 2 3 LENGTH CODES OVER GR (5
;
2)
(a) Cyclotomic coset representative set L .
(b) Constrained sets die
(c) Constrained sets die
Similarly, if D = fD1 ; D2 ; . . . ; Du g is a constraint for a Q -invariant -consta-Abelian code defined in Theorem 2, it is straightforward to check that, for any Q -invariant -consta-Abelian constrained
set Dj , the corresponding set Dj? , which is a collection of representative elements of the cyclotomic coset containing k? , for all k 2 Dj , is
a valid constrained set for a Q -invariant 01 -consta-Abelian code.
We call such a pair of constrained sets Dj and Dj? a dual constrained
set pair.
Theorem 5: If (T0 ; T1 ; . . . ; Ta ) is the defining partition of a n =
mr01 mr02 1 1 1 m0 length Q -invariant (resp., U -invariant)
-consta-Abelian code C over GR (pa ; l) and if the set Dj T ,
then its dual code is a Q -invariant (resp., U -invariant)
01 -consta-Abelian code with defining partition (T0? ; T1? ; . . . ; Ta? )
such that the corresponding dual constrained set Dj? Ta?0 .
Proof: For both Q -invariant and U -invariant -consta-
Abelian code, since we have already shown that Dj? is a valid constrained set for 01 -consta-Abelian code, the result follows directly
from Theorem 4 and Theorem 2 (resp., Theorem 3).
Self-dual -consta-Abelian codes can be discussed only if = 01
which implies self-dual -consta-Abelian codes exist only for , with
?
= 1 or 01 for all . If = 1, then h = h = 0, d = 1 and
?
if = 01, h = h = 1 and d = 2. Two special cases are as
follows.
i) = (1; 1; . . . ; 1) which correspond to the Abelian codes over
GR (pa ; l). The dual-cyclotomic coset of die reduces to the coset
dn 9 ie as discussed in [14].
ii) = (01; 01; . . . ; 01) which are nega-Abelian codes (generalization of negacyclic) over GR (pa ; l). The dual-cyclotomic coset
of die is the coset containing
d01 0 ir01; 01 0 ir02; . . . ; 01 0 i0e:
From Theorem 5, a Q -invariant (resp., U -invariant) -constaAbelian code with defining partition (T0 ; T1 ; . . . ; Ta ) is self-dual if
and only if whenever Dj T , the corresponding dual constrained set
for different values of s.
for different values of b.
is a subset of Ta0 . We have a nonexistence result for self-dual codes
which is an extension of a similar result for Abelian codes.
Theorem 6: If gcd(p; n) = 1, a is an odd integer and is
such that = 01 for all 2 S Ir , = 1 otherwise, then
n = mr01 mr02 1 1 1 m0 length self-dual -consta-Abelian codes over
GR (pa ; l) do not exist if 2S m is an odd integer.
Proof: Consider the element i = dir01 ; ir02 ; . . . ; i0 e 2 In such
that i = m 201 for all 2 S and i = 0 for all 2 Ir n S . The dual
m 01
?
?
= i for all
element is i? = di?
r01 ; . . . ; i0 e with i = 01 0
2
?
2 S and i = 0 = i for all 2 Ir n S . Hence, the dual cyclotomic
coset of die is die itself.
For the self-dual code C , if Ai takes values from an -ideal, then
according to Theorem 5, the corresponding dual transform component
which is Ai itself, should take values from (a 0 )-ideal. This implies
a 0 = ) a = 2 , and hence a must be even.
Corollary 2: If gcd(p; n) = 1 and n = mr01 mr02 1 1 1 m0 is an
odd integer, self-dual nega-Abelian codes of length n over GR(pa ; l)
do not exist for odd values of a.
VI. CONCLUSION
In this correspondence, using a TDFT, we have characterized
consta-Abelian codes invariant under two monomials Q and U .
Codes invariant under Q generalize the class of QT codes which are
known to contain some good codes. Consta-Abelian codes are also
( r01 ; nr01 )-QT, and they have an advantage over ( r01 ; nr01 )-QT
codes that are not consta-Abelian, because they need a smaller
extension ring for DFT characterization. We have also shown that the
dual of a Q -invariant (resp., U -invariant) -consta-Abelian code
is a Q -invariant (resp., U -invariant) 01 -consta-Abelian code.
In [18], it was shown that certain Abelian codes over a Galois ring
are isomorphic to a direct sum of consta-Abelian codes. For such
Abelian codes, knowledge of the automorphism group of individual
consta-Abelian codes give valuable information about the automorphism group of the Abelian code itself. Thus, for such Abelian codes,
the results of this correspondence can be used to get subgroups of
automorphism group which were not considered in our earlier paper
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
[14]. Further, encoding and decoding of such Abelian codes can be
efficiently performed in the DFT domain by using parallel TDFTs (one
for each consta-Abelian code) instead of using a single generalized
DFT for Abelian codes used in [14].
APPENDIX
THEOREM 2 FOR THE PARTICULAR CASE s =
=
0
(
i=0
=0
j
r 1 r 1
0 )
0
(
0
r 1
j
r 2 r 2
0
0 )
i
=0
ai :
i j
Using inverse TDFT expression
0
r 1
ai = n01
=0
0i
n 1
0
r 1
0
j =0
=0
0i
Aj
j
we can further simplify the preceding equation, as shown at the bottom
of the page and finally obtain (20), where
K=
1
mr02
(
0 1)
j
r 1 r 1
0
0
k
r 2 r 2
0 Ak :
0
k2J
But, from Lemma 2, the latter condition is possible if and only if Ak =
0 for all Ak 2 J (r02) (j ), which means j is not free. Therefore, j is
free if and only if
(j )
Note that, in the equations at the bottom of this page, the second
(r 02)
equality is reduced to the third equality using the fact that r01 = 1
(r 02)
if ir02 01 mod mr02 and r01 = 0 for other values of ir02 .
Now, j 2 L is free (the values of Aj does not influence the values
that other transform components can take and vice versa) if and only if
K = 0, which is true if and only if r01 jr01 = 1 or
j
r 1 r 1
0 0 =1
hr01 + dr01 jr01 0 mod mr01 dr01
dr01 j hr01 and jr01 0 hr01 mod mr01
dr01
m
h
m
=
=
=
1
r01
r01
r01
< r 0 1; hr01 = 0; dr01 = 1 and jr01 = 0hr01 = 0:
()
()
()
()
r02
For any n and p such that gcd(n; p) = 1, a length n =
mr01 mr02 1 1 1 m0 -consta-Abelian code over GR(pa ; l) is
Q -invariant for = (1; . . . ; 1; 6= 1; . . . ; 0 ) if and
only if it satisfies the constraint D = fD1 ; D2 ; . . . ; Du g, where
j 2 Di L ) Di = dj e<r02> .
Proof: Let ~a 2 C . For the -consta-Abelian code C to be
Q -invariant, Q (~a) 2 C . Let
A~ (r02) = (A(0r02) ; A(1r02) ; . . . ; A(nr0012) )
be the TDFT vector of Q (~a). From the TDFT expression, we have
n01 r01
i
i j
Q(i) (ai0n )
A(jr02) =
i=0
n 1
379
To prove the other cases,
((): Assume that the -consta-Abelian code C satisfies the given
constraint. That is, for all i 2 dj e<r02> , Ai takes values from their
respective -ideal (C i = p GR (pa ; lei )). This implies all Ak ,
k 2 J (r02) (j ) take values from their -ideal (see Definition 9), and,
therefore, K is definitely an element of the ideal p GR (pa ; lm).
le
It is straightforward to check that 0 (K ) = K (which implies
a
K 2 p GR (p ; lej )) and hence,
0le (A(jr02) ) = 0le ( r02 jr02 Aj + K ) = A(jr02) :
This means that if C is a -consta-Abelian code with C j =
p GR (pa ; lej ), then C j (r02) = p GR (pa ; lej ) for any j . Therefore, C is Q -invariant.
()): Let the -consta-Abelian code C be Q -invariant, with
C j = p GR (pa ; lej ) for j 2 L . We need to prove that all other spectral components in the set dj ehr02i take values from the -ideal of their
corresponding Galois subrings. It is enough to prove this for the spectral components in J (r02) (j ) since the other components in dj ehr02i
will get connected through conjugacy constraints. Toward this end, we
consider another transform component j 0 such that j 0 2 J (r02) (j ) and
C j = p GR (pa ; lej ). All we need to show is that -consta-Abelian
code C is Q -invariant if and only if = 0 .
From (20)
K = A(jr02) 0 jr02 Aj :
Since Aj takes values from the ideal p GR (pa ; lej ) and the code is
Q -invariant, A(jr02) also takes values from p GR (pa ; lej ) and,
j
hence, we have K 2 p GR (pa ; lej ). Since m 1 ( r01 0 1) is a
unit, this implies
k
r 2 r 2
0 Ak = 0:
0
2
k J
(j )
2
k J
(j )
n01 n01
r01
j
j
A(jr02) = 1
( r01 r01 )
( r02 r02 )
( )i (j 0k ) Ak
n k=0 i=0
=0
n01
j
r02
=
( r02 )i (j 0k )+j ( r01 r01 )
n
i
=0
k2J
(j )
=
=
=
0
n
2
k J
(j )
j
r 2 r 2
0
0
i=0;
0 Aj + K
0k
i
(j
r 2
0
)
0
i=0;
0 1)
2
k J
(21)
Ak
n 1
Ak +
=0
1
j
0 Aj + mr02 ( r01 r01
j
r 2 r 2
0
0
n 1
j
r 2
r 2
a
0 Ak 2 p GR (p ; lej ):
k
r 2
k
r 1 r 2
0
0
j
r 1
0 Ak
=1
k
r 2 r 2
0
(j )
0 Ak
(20)
380
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 2, FEBRUARY 2004
The transform component j 0 2 J (r02) (j ) and, from Lemma 2, (21)
implies 0 .
Notice that j 2 J (r02) (j 0 ) = J (r02) (j ) and, hence, in the counter(r 02)
, K is a constant, i.e.,
part of (20) for Aj
A(jr02) =
j
r02
Aj
+ K:
By a similar argument to that which we used to obtain (21), we find
that
k2J
which implies (j )
k
r02
Ak 2 p
[22] B. K. Dey and B. S. Rajan, “F –linear cyclic codes over F : DFT
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S. Boztas and I. Shparlinski, Eds. Berlin, Germany: Springer-Verlag,
2001, vol. 2227, pp. 67–76.
[23] S. Boztas, “Constacyclic codes and constacyclic DFTs,” in Proc. IEEE
Int. Symp. Information Theory, Cambridge, MA, June 1998, p. 235.
[24] W. C. Huffman, “Decompositions and extremal type II codes over Z ,”
IEEE Trans. Inform. Theory, vol. 44, pp. 800–809, Mar. 1998.
GR (pa ; lej )
0 . Hence, = 0 .
ACKNOWLEDGMENT
The authors are grateful to the anonymous reviewers for their helpful
comments.
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