A Geometry of Coding Theory
PARTHA PRATIM DEY
Department of Computer Science
North South University
12 Kemal Ataturk Rd., Banani, Dhaka
BANGLADESH
Abstract: - The connection between projective planes and quantum mechanics is well known [1]. In this paper, we
explore the connection of finite projective planes with the theory of coding. The focus is the dimension of the linear
codes that are obtained from the incidence systems of projective planes. A good many valuable features of a linear
code like its size, number of data and check bits in a codeword, the transmission rate and dimensions of encoder
and decoder matrices are obtained when the dimension of the code is known. Compact disk players [2], hard disk
drives [3], high-speed modems [4] are some of the components of classical computers, which make use of some of
these linear codes to improve reliability of transmission of information over some noisy channel. The idea of
quantum information processing on the other hand is inconceivable [5] without the error-correction facilities of
these codes or their quantum adaptations.
Key-Words:- Projective planes, dimension, cartesian pair, incidence matrix, codes, centralizers.
1 Introduction
After Euclidean geometry, one of the first to become
popular was projective geometry. It was the painters
of Italian Renaissance who initiated the study of this
geometry [6]. Finite projective planes came from
two-dimensional projective geometry.
Suppose F is any field. The space of all vectors
(a 0 ,..., a n ), a i  F is called the projective geometry
of dimension n over F , denoted PG(n, F ) . The
zero vector 0  (0,...,0) is the void space, and we say
that the void space has dimension  1 . A point P , of
dimension 0, is the set of vectors
bx  (bx 0 ,..., bx n )
where x  ( x 0 ,..., x n )  0 and b ranges over all
elements of F . More generally if y 0 ,..., y k are
k  1 independent vectors, the set of all vectors
b0 y 0  ...  bk y k , bi  F is a subspace S k of
dimension k . A subspace of dimension n  1 is
called a hyperplane. If (c 0 ,..., c n )  (0,...,0), it can
be
shown that the
( x0 ,..., x n ) satisfying
c 0 x 0  ...  c n x n  0
set
of
all
vectors
is a hyperplane, and conversely, that every
hyperplane can be defined this way, (c 0 ,..., c n ) and
(sc 0 ,..., sc n ), s  0, defining the same point.
If the field F is the field GF ( p r ), then writing
there
are
q  pr ,
( x0 ,..., x n ), x i  GF ( p r ) .
q n 1
Each
vectors
of
the
n 1
 1 vectors different from zero determines a
( x0 ,..., x n ) and
point,
and
since
(bx 0 ,..., bx n ), b  0 , determine the same point, there
q
q n 1  1
points. In the same way, since
q 1
(c0 ,..., c n )  (0,...,0) and
(sc 0 ,..., sc n ), s  0 ,
determine the same hyperplane, there are v
hyperplanes. If
y 0 ,..., y t are t  1independent
are v 
vectors, then b0 y 0  ...  bt y t , bi  GF ( p r ) gives
q t 1 vectors. Excluding the zero vector, we have
q t 1  1
different points in the space S t of
q 1
t.
dimension
Thus
a
hyperplane
has
n
q 1
k
different points. Singer [7] has proved
q 1
that the hyperplanes of PG(n, p r ) , q  p r as blocks,
points
as
objects,
form
a
symmetric
(v, k ,  )  design with
q n 1  1
qn 1
q n 1  1
and  
.
v
,k
q 1
q 1
q 1
If n  2, then
v  q 2  q  1, k  q  1 and
This
special 2  1.
2
dimensional (q  q  1, q  1,1) -symmetric design is
called a projective plane of order q .
However there is another approach to projective
plane that is far more general than the approach
above. In this approach, a projective plane of
order n is
an
incidence
structure
2
with n  n  1 points and lines. Any two distinct
points are contained on exactly one line and two
distinct lines intersect in a unique point. Each line
consists of n  1 points and each point belongs to
precisely n  1 lines. For each line L of the plane  ,
we may form the derived design whose points are
the n 2 points of the set   L and whose lines are
the n 2  n lines obtained by deleting the points
of L from each line of  . This design, denoted by
  L , is called the affine plane afforded by L . Hall
has shown [8], it is possible to coordinatize any
projective plane by means of a ternary ring. When the
ternary ring is a field, the plane is called
Desarguesian.
lines so that there are n line orbits of length n and
exactly n  1 fixed lines.
Let us now assume that  is a plane of order n
and G = {g1  1,..., g n } is a Cartesian group with
axis L and center C . Let L1 ,..., Ln denote the n lines
of  incident with C and different from L . Set
X i  Li \ {C} and choose a base point xi  X i .
Then
as
G acts
on X i , X i  {xi g1 ,..., xi g n } and
X 1 ,..., X n partition the n 2 points of the affine plane
 \ {L} . Likewise, if (1),..., (n) denote the points of
L different from C , we may pick a base line l m  L
so that Ym  {l m g1 ,..., l m g n } is the set of lines of
 different from L and incident with (m) . The sets
Y1 ,..., Yn partition the n 2 lines of  which do not
contain C and we define hmi to be the
n  n incidence matrix for the pair (Ym , X i ) . That is,
hmi has (r , s )  entry one if the point xi g s is
incident with the line l m g r . Because the lines of
Ym correspond to a parallel class of the affine plane,
hmi is a permutation matrix and the n 2  n 2 matrix
H  ( hij ) is an incidence matrix for the n 2 lines and
points not fixed by G .
We may enlarge
2 Cartesian Connection of the Incidence Matrix
A Cartesian group G of order n is an
automorphism group of  with n elements called
elations. Each nontrivial elation of G fixes
pointwise a distinguished line L (called the axis of
G ) and fixes no other point. Dually, L contains a
distinguished point C (called the center of G ) and
each element of G fixes the n  1 lines incident with
C and no further lines. The group G is called a
(C , L )  transitivity as G transitively permutes the
n points of M \ {C} for any line M containing
C , M  L . This implies that there are precisely n
point orbits for G of length n and exactly n  1
singleton point orbits. Likewise G permutes the
transitively
the
sets
H  ( hij ) to an incidence
matrix A of  by the addition of n  1 more
columns and rows with the last n  1 columns
corresponding the points (1),..., ( n), C of L and the
last n  1 rows labeled by the lines L1 ,..., Ln , L . We
obtain a matrix,
H B t 
(1) A  
 , H  ( hij )
B M 
where
B, M are rectangular matrices of size
(n  1)  n 2 and (n  1)  (n  1) respectively. Note
that the first n rows of B are the characteristic
vectors for the sets X i and independent of the base
point chosen. Each entry in the last row and column
of M is one and all other entries are zero.
Hamming codes, BCH codes etc. are examples of
such codes.
The dimension of the code of a Desarguesian
projective plane of order q  p r is equal to
 p  1

  1 .
2


r
Theorem ( 2.1) [9]. Assume G is a Cartesian group of
order n and A is the incidence matrix (1) . Then
H  ( hij ) is a matrix of order n whose entries
hij belong to the regular permutation representation
R (G ) . Moreover, for m  k the sequence
{hmi hki1 | i  1,..., n}
contains each element of R (G ) exactly once.
This formula for rank was conjectured by Rudolph
and has been proved, using different methods , by
Goethals and Delsarte [10], by MacWilliams and
Mann [11] and finally by Smith [12].
But in the case of non-Desarguesian planes we do
not have similar formulas. The only known result
[13]states that if  is a plane of order n , C its code
over F p and p divides n exactly to the first power,
then
The matrix H in Theorem ( 2.1) is called a
generalized Hadamard matrix of order n for R (G ) .
Since G  R (G ) , we may identify the elements of
R (G ) with those of G so that H  ( hij ) is also a
generalized Hadamard matrix for G of order n . We
call G a Cartesian group and (G , H ) a Cartesian pair.
3 Codes of the Planes
Let F p denote the finite field with p elements, and
let F pv denote
the
set
tuples ( x1 ,..., xv ), where xi  F p .
of v A
[v, s ] code C over F p is a linear subspace of F pv of
dimension s . The vectors of C are called codewords,
and sometimes more briefly words. The dual
code C  is the subspace orthogonal to C under the
usual scalar product on V . That is

C  {x V | (u, x)  0 for all u  C}
Here C  is a [v, v  s ] code.
Definition (3.1) . The code C of a plane  of order n is
defined as the linear subspace of F pv spanned by the
rows of its incidence matrix considered as vectors
over F p where v  n 2  n  1 is the number of points
(lines) in  .
n2  n  2
n2  n
and dim C  
2
2

and C
is a subcode of C of codimension
1. However, there is a similar theorem [14] for the
code of an affine plane. It states that if C is the code
of an affine plane   L of order n , then
n2  n
dim C 
2
when p divides n exactly to the first power.
Moreover if  affords a C  L transitivity, then C is
a G -invariant subcode. We have been able [15] to
prove that if C is a code over F p and H is a group of
dim C 
linear transformations leaving C invariant, then
q 1
dim C   dim CC (hi )  q dim CC ( H )
i 1
where H  Z q2 with q  p , h1 ,..., hq 1 are
q  1 subgroups
and CC (hi ), CC ( H ) are
of
of
generators
H order
centralizers
q,
of
hi and H respectively.
VanAken has been able to determine the
dimension of the code of a biaffine plane under the
assumption that p divides n exactly to the first
power. A biaffine plane of order n is an incidence
system which is obtained by deleting just one point
from an affine plane.
Theorem (VanAken). Let D be an affine difference
set of order n in a group G , G  n 2  1. If p is a
prime divisor of n and p 2 does not divide n , the
code C
over F p of the development of D
dimension
n2  n  2
.
2
has
The above theorem fails completely if p 2 | n . In
general the dimension of C ext is much smaller than
n2  n  2
C ext of
.
The
dimension
of
2
PG(2,2 r ), for example, has been computed exactly
by the coding theorists. According to this
computation, if the order of the plane is 2 r , then
We conclude this section with a brief discussion
on the extended code C ext of a projective plane  of
order n . The extended code of  is the F p  span of
[length,
the rows of the extended incidence matrix B given as
follows.
References:
A

B


1...1





n  1
1

1
where A is the incidence matrix of  . Instead of
ordinary dot product we define the bilinear form
 by
 ( x, y)  x1 y1  ...  xv y v  xv 1 y v 1
for
x  ( x1 ,..., xv 1 ) and
y  ( y1 ,..., y v 1 ) with
v  n 2  n  1 . This form  is called the Minkowski
product. One checks that if x and y are rows of
B , then
 ( x, y )  0 or
Thus
n.
 ( x, y )  0 (mod p) . The extended code C ext is
thus self orthogonal with respect to the Minkowskii
product. In fact the following theorem states that
when p divides n exactly to the first power, C ext is
actually self-dual.
Theorem (3.2) [16] Let  be a projective plane of
order n . If p divides n exactly to the first power,
then the extended code of  has dimension
n2  n  2
and is self-dual with respect to the
2
Minkowskii
 ( x, y)  x1 y1  ...  xv y v  xv 1 y v 1 .
product
[2
2r
dimension]
r
 2  2,3 r  1] .
of C ext is
given
by
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