SEMINAR: Groups and Combinatorics Seminar, Designs and Elusive Codes in Hamming Graphs

Daniel Hawtin (UWA)

will speak on

Designs and Elusive Codes in Hamming Graphs

at 3pm Friday August the 22nd in Weatherburn Lecture Theatre.

Abstract:

We consider a code to be a subset of the vertices of a Hamming graph. The set of s-neighbours of a code are those vertices which are distance s from some codeword, but not distance r from any codeword, for 0 <= r < s. The automorphism group of a code necessarily fixes the set of s-neighbours of the code. An s-elusive code is a code such that the automorphism group of the set of s-neighbours is strictly larger than the automorphism group of the code itself. We provide examples for s=1,2,3, including a family of Reed-Muller codes and the Preparata codes. We also discuss some restrictions on the parameters of elusive codes and show that an elusive code gives rise to a q-ary t-design. This fact is used to provide a new construction for the Nordstrom-Robinson code.