Double circulant self-dual and LCD codes over Galois rings

TL;DR

This paper explores the existence, construction, and asymptotic properties of self-dual and LCD double circulant codes over Galois rings, providing new algorithms and demonstrating their asymptotic goodness.

Contribution

It introduces an algorithm for a duality-preserving Gray map over Galois rings and constructs asymptotically good self-dual and LCD codes over rac{p^2}{}, expanding coding theory over rings.

Findings

Existence and enumeration results for these codes.

An explicit Gray map construction for p rac{p _{p^2}}.

Families of asymptotically good codes via random coding.

Abstract

This paper investigates the existence, enumeration and asymptotic performance of self-dual and LCD double circulant codes over Galois rings of characteristic $p^2$ and order $p^4$ with $p$ and odd prime. When $p \equiv 3 \pmod{4},$ we give an algorithm to construct a duality preserving bijective Gray map from such a Galois ring to $\mathbb{Z}_{p^2}^2.$ Using random coding, we obtain families of asymptotically good self-dual and LCD codes over $\mathbb{Z}_{p^2},$ for the metric induced by the standard $\mathbb{F}_p$-valued Gray maps.