Generalised rank weights of quasi-cyclic codes

Abstract

Generalised rank weights are a set of parameters characterising the performance of a linear code over Fqm in secure network coding. In this report we study the generalised rank weights of quasi-cyclic codes, a special class of linear codes usually studied for their properties in error correction over the classical Hamming metric. By using the algebraic structure of quasi-cyclic codes, a new upper bound on the generalised rank weights of quasi-cyclic codes is formulated. Additionally, it is shown that the first generalised rank weight of self-dual 1-generator quasi-cyclic codes is almost completely determined by the choice of Fqm.