## Linear codes obtained from 2-modular representations of some finite simple groups.

##### Abstract

Let F be a finite field of q elements and G be a primitive group on a finite set
. Then
there is a G-action on
, namely a map G
!
, (g; !) 7! !g = g!; satisfying
!gg0 = (gg0)! = g(g0!) for all g; g0 2 G and all ! 2
, and that !1 = 1! = !
for all ! 2
: Let F
= ff j f :
! Fg, be the vector space over F with basis
. Extending the G-action on
linearly, F
becomes an FG-module called an FG-
permutation module. We are interested in finding all G-invariant FG-submodules,
i.e., codes in F
. The elements f 2 F
are written in the form f =
P
!2
a! !
where ! is a characteristic function. The natural action of an element g 2 G is
given by g
P
!2
a! !
=
P
!2
a! g(!): This action of G preserves the natural
bilinear form defined by
*
X
a! !;
X
b! !
+
=
X
a!b!:
In this thesis a program is proposed on how to determine codes with given
primitive permutation group. The approach is modular representation theoretic and
based on a study of maximal submodules of permutation modules F
defined by
the action of a finite group G on G-sets
= G=Gx. This approach provides the
advantage of an explicit basis for the code. There appear slightly different concepts
of (linear) codes in the literature. Following Knapp and Schmid [83] a code over
some finite field F will be a triple (V;
; F), where V = F
is a free FG-module of
finite rank with basis
and a submodule C. By convention we call C a code having
ambient space V and ambient basis
. F is the alphabet of the code C, the degree
n of V its length, and C is an [n; k]-code if C is a free module of dimension k.
In this thesis we have surveyed some known methods of constructing codes from
primitive permutation representations of finite groups. Generally, our program is
more inclusive than these methods as the codes obtained using our approach include
the codes obtained using these other methods. The designs obtained by other authors
(see for example [40]) are found using our method, and these are in general defined
by the support of the codewords of given weight in the codes. Moreover, this method
allows for a geometric interpretation of many classes of codewords, and helps establish
links with other combinatorial structures, such as designs and graphs.
To illustrate the program we determine all 2-modular codes that admit the
two known non-isomorphic simple linear groups of order 20160, namely L3(4) and
L4(2) = A8. In the process we enumerate and classify all codes preserved by such
groups, and provide the lattice of submodules for the corresponding permutation
modules. It turns out that there are no self-orthogonal or self-dual codes invariant
under these groups, and also that the automorphism groups of their respective codes
are in most cases not the prescribed groups. We make use of the Assmus Matson
Theorem and the Mac Williams identities in the study of the dual codes. We observe
that in all cases the sets of several classes of non-trivial codewords are stabilized
by maximal subgroups of the automorphism groups of the codes. The study of
the codes invariant under the simple linear group L4(2) leads as a by-product to a
unique
flag-transitive, point primitive symmetric 2-(64; 28; 12) design preserved by
the affi ne group of type 26:S6(2). This has consequently prompted the study of binary
codes from the row span of the adjacency matrices of a class of 46 non-isomorphic
symmetric 2-(64; 28; 12) designs invariant under the Frobenius group of order 21.
Codes obtained from the orbit matrices of these designs have also been studied.
The thesis concludes with a discussion of codes that are left invariant by the simple
symplectic group S6(2) in all its 2-modular primitive permutation representations.