Structure of Cayley Codes

Abstract

Cayley codes, introduced by Kaufman and Wigderson, are linear codes constructed from a Cayley graph and a smaller linear code. We explore general properties of the class of Cayley codes for finite groups. In particular we give a reduction to Cayley codes for connected Cayley graphs that maintains code properties such as rate, minimum distance and symmetry. Also, for a given Cayley code, we identify a family of symmetric Cayley codes, each associated with a normal edge-transitive Cayley graph, such that the given Cayley code embeds into the direct sum of the symmetric Cayley codes. We analyse several families of examples, in particular studying the behaviour of the Cayley code construction under forming direct products and cartesian products of Cayley graphs, and we pose a number of open questions.