Double circulant self-dual and LCD codes over Galois rings

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 $p\equiv 3(\mathrm{mod}4),$ 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.

Sbumitted on 4, December, 20 pages