On self-dual negacirculant codes of index two and four

TL;DR

This paper investigates the structure and enumeration of self-dual negacirculant codes of index two and four over finite fields, focusing on cases where the code length relates to primes congruent to 3 mod 4, and explores their properties using number theory conjectures.

Contribution

It provides a detailed analysis of self-dual negacirculant codes of specific lengths, including enumeration and bounds on their relative distance, under certain number-theoretic conditions.

Findings

Exact enumeration of self-dual negacirculant codes

Modified Varshamov-Gilbert bound established

Infinitely many primes p satisfy conditions under GRH

Abstract

In this paper, we study a special kind of factorization of $x^n+1$ over $\mathbb{F}_q, $ with $q$ a prime power $\equiv 3~({\rm mod}~4)$ when $n=2p,$ with $p\equiv 3~({\rm mod}~4)$ 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.