Self-Dual Doubly Even $2$-Quasi-Cyclic Transitive Codes Are Asymptotically Good

Abstract

In this correspondence, we prove that the class of binary self-dual doubly even 2-quasi-cyclic transitive codes is asymptotically good. This improves a recent result of Bazzi and Mitter ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IEEE Trans. Inf. Theory</i> , vol. 52, pp. 3210-3219, 2006). The proof is based on the study of a particular class of codes invariant under dihedral groups using a blend of representation theory and probabilistic arguments. The methods are closely related to those used in Bazzi and Mitter. In order to complete the proof a number theoretical result of Hasse is needed.