Constructing designs and codes from fixed points of alternating groups

Abstract

The study of finite structures in discrete mathematics is a broad area which has many in influential results not only in mathematics but also in practice. It is well known that 1􀀀designs have many applications in coding theory. The construction of these designs and codes from xed points of alternating groups plays a central role in obtaining properties of codes, structures and other general results that are useful in application of coding theory. In this dissertation we construct some 1􀀀designs from (Key-Moori method 2) and J Moori Method 3. In (Key-Moori method 2), we have a technique from which a large number of nonsymmetric 1􀀀designs could be constructed from maximal subgroups and conjugacy classes of elements of nite groups. In this dissertation, we consider the alternating group G = An with its maximal subgroup M = An􀀀1 where nX is a conjugacy class of elements of order n in G. Let g 2 nX, then jCgj = j[g]j = [G : CG(g)]. Using J Moori Method 3, we construct some 1􀀀designs from the xed points of elements of alternating groups. While J Moori has accomplished most of the work by constructing all designs and codes for alternating groups, and A Saeidi constructed designs and codes from involutions of alternating groups An and for maximamal subgoup isomorphic to Sn􀀀2. In this dissertation we will be looking at other maximal subgroups (particularly the maximal subgroup An􀀀1 of An) which are not covered by J Moori and A Saeidi. Therefore, making this work an extention to J Moori and A Saeidi work.