Monday, Mar 2, 2015 - 3:30pm - Allen 19
Top Combo Fun seminar
Automorphism groups of codes I
Dr. Ted Dobson, Mathematics, Msstate
Title: Automorphism groups of codes
Abstract: In a recent arXiv posting, Mikhail Muzychuk noticed a relationship between the isomorphism problem for Cayley digraphs of a group G and the isomorphism problem for codes permutation invariant under G. For cyclic groups, he showed that in fact the permutation isomorphism problem for cyclic codes reduces to the isomorphism problem for circulant digraphs. This latter problem has been completely solved, and so Muzychuk produced a solution to the permutation isomorphism problem for cyclic codes. We consider the problem of computing the automorphism group of cyclic codes (and codes invariant under other groups as well). We first give a sufficient condition to decompose a code C into two subcodes C1 and C2, both invariant under the permutation automorphism group of C, and which are determined by codes of smaller length. Additionally, we show that PAut(C) = PAut(C1) ∩ PAut(C2). This sufficient condition corresponds to an existing sufficient condition that gives a similar decomposition of a vertex-transitive digraph. I will give two seminars, with the first discussing the background of the Cayley isomorphism problem for graphs and discussing the proof of the decomposition theorem for a vertex-transitive digraph. The second will give basic information about codes, and then discuss the decomposition theorem for codes. This is joint work with Mikhail Muzychuk of Netanya Academic College.