Weight Distributions of Codes Over Finite Rings

Speaker: Dr. Eimear Byrne

Time: 4:00PM

Date: Mon 16th April 2012

Location: Mathematical Sciences Seminar Room (Ag 1.01)

Let R > S be finite rings for which there exists a trace map T from R onto S, that is an S-linear epimorphism from R onto S whose kernel contains no nontrivial ideal of R. In the case that S has a generating character (is a Frobenius ring) X, the composition of X with T yields a generating character for R.

Given a function f : R -> R we define an S-linear code whose words are the distinct functions T(ax+bf(x)) for a,b in R. The (homogeneous) weight of a word can be expressed in terms of a character sum or generalized Walsh tranform. Such functions f and their corresponding codes have been studied extensively for finite fields and Galois rings. There are connections to symmetric cryptography, to sequences and to combinatorics. In this talk we consider functions on integer modular rings and commutative local Frobenius and compute the weight distributions of the corresponding codes.

(This talk is part of the Algebra/Claude Shannon Institute series.)