On self-dual negacirculant codes of index two and four

Abstract

In this paper, we study a special kind of factorization of $x^n+1$ over ${\mathbb{F}}_q,$ with $q$ a prime power $\equiv 3(\mathrm{mod}4)$ when $n=2p,$ with $p\equiv 3(\mathrm{mod}4)$ and $p$ is a prime. Given such a $q$ infinitely many such $p$'s exist that admit $q$ as a primitive root by the Artin conjecture in arithmetic progressions. This number theory conjecture is known to hold under GRH. We study the double (resp. four)-negacirculant codes over finite fields ${\mathbb{F}}_q,$ of co-index such $n$'s, including the exact enumeration of the self-dual subclass, and a modified Varshamov-Gilbert bound on the relative distance of the codes it contains.

Design, Codes and Cryptography,2018

Cited by in corpus (1)

• Quaternary Hermitian self-dual codes of lengths 26, 32, 36, 38 and 40 from modifications of well-known circulant constructions