Discrete mathematics has had many applications in recent years and this is
only one reason for its increasing dynamism. The study of finite structures is
a broad area which has a unity not merely of description but also in practice,
since many of the structures studied give results which can be applied to other, apparently dissimilar structures. Apart from the applications, which themselves generate problems, internally there are still many difficult and interesting problems in finite geometry and combinatorics. There are still many puzzling features about sub-structures of finite projective spaces, the minimum weight of the dual codes of polynomial codes, as well as about finite projective planes. Finite groups are an ever strong theme for several reasons. There is still much work to be done to give a clear geometric identification of the finite simple groups. There are also many problems in characterizing structures which either have a particular group acting on them or which have some degree of symmetry from a group action.
Codes obtained from permutation representations of finite groups have been given particular attention in recent years. Given a representation of group elements of a group G by permutations we can work modulo 2 and obtain a representation of G on a vector space V over lF2 . The invariant subspaces (the subspaces of V taken into themselves by every group element) are then all the binary codes C for which G is a subgroup of Aut(C). Similar methods produce codes over arbitrary fields. Through a module-theoretic approach, and based on a study of monomial actions and projective representations, codes with given transitive permutation group were determined by various authors. Starting with well known simple groups and defining designs and codes through the primitive actions of the groups will give structures that have this group in their automorphism groups. For each of the primitive representations, we construct the permutation group and form the orbits of the stabilizer of a point.
Taking these ideas further we have investigated the codes from the primitive permutation representations of the simple alternating and symplectic groups of odd characteristic in their natural rank-3 primitive actions. We have also investigated alternative ways of constructing these codes, and these have come about by noticing that the codes constructed from the primitive permutations of the groups could also be obtained from graphs. We achieved this by constructing codes from the span of adjacency matrices of graphs. In particular we have constructed codes from the triangular graphs and from the graphs on triples.
The simple symplectic group PSp2m(q), where m is at least 2 and q is any prime power, acts as a primitive rank-3 group of degree q2m-1/q-1 on the points of the projective (2m-1)-space PG2m-1(IFq ). The codes obtained from the primitive rank-3 action of the simple projective symplectic groups PSp2m(Q), where Q= 2t with t an integer such that t ≥ 1, are the well known binary subcodes of the
projective generalized Reed-Muller codes. However, by looking at the simple symplectic groups PSp2m(q), where q is a power of an odd prime and m ≥ 2, we observe that in their rank-3 action as primitive groups of degree q2m-1/q-1 these groups have 2-modular representations that
give rise to self-orthogonal binary codes whose properties can be linked to those of
the underlying geometry. We establish some properties of these codes, including
bounds for the minimum weight and the nature of some classes of codewords.
The knowledge of the structures of the automorphism groups has played a key
role in the determination of explicit permutation decoding sets (PD-sets) for the
binary codes obtained from the adjacency matrix of the triangular graph T(n) for n ≥ 5 and similarly from the adjacency matrices of the graphs on triples.
The successful decoding came about by ordering the points in such a way that the
nature of the information symbols was known and the action of the automorphism
group apparent.
Although the binary codes of the triangular graph T(n) were known, we have
examined the codes and their duals further by looking at the question of minimum weight generators for the codes and for their duals. In this way we find bases
of minimum weight codewords for such codes. We have also obtained explicit
permutation-decoding sets for these codes.
For a set Ω of size n and Ω{3} the set of subsets of Ω of size 3, we investigate the binary codes obtained from the adjacency matrix of each of the three graphs with
vertex set Ω{3}1 with adjacency defined by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively. We show that
permutation decoding can be used, by finding PD-sets, for some of the binary codes obtained from the adjacency matrix of the graphs on (n3) vertices, for n ≥ 7. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 2002.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:ukzn/oai:http://researchspace.ukzn.ac.za:10413/3895 |
| Date | January 2002 |
| Creators | Rodrigues, Bernardo Gabriel. |
| Contributors | Moori, Jamshid., Key, Jennifer Denise. |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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