On self-dual and LCD double circulant and double negacirculant codes over $\mathbb{F}_q + u\mathbb{F}_q$

TL;DR

This paper investigates the structure, enumeration, and construction of self-dual and LCD double circulant and negacirculant codes over a specific ring, providing bounds on their parameters and examples of optimal codes.

Contribution

It provides the first exact enumeration of self-dual and LCD double circulant and negacirculant codes over the ring and constructs new codes over finite fields using Gray maps.

Findings

Exact enumeration of self-dual and LCD codes for given lengths.

Construction of new self-dual and LCD codes over finite fields.

Several constructed codes are optimal with good parameters.

Abstract

Double circulant codes of length $2n$ over the semilocal ring $R = \mathbb{F}_q + u\mathbb{F}_q,\, u^2=u,$ are studied when $q$ is an odd prime power, and $-1$ is a square in $\mathbb{F}_q.$ Double negacirculant codes of length $2n$ are studied over $R$ when $n$ is even and $q$ is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length $2n$ is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length $4n$ over $\mathbb{F}_q$ are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below. The parameters of examples of the modest length are computed. Several such codes are optimal.