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 $p$ an odd prime. When $p \equiv 3 \ ({\rm mod} \ 4),$ we give a method to construct a duality preserving bijective Gray map from such a Galois ring to $\mathbb{Z}_{p^2}^2.$ Closed formed enumeration formulas for double circulant codes that are self-dual (resp. LCD) are derived as a function of the length of these codes. Using random coding, we obtain families of asymptotically good self-dual and LCD codes over $\mathbb{Z}_{p^2}$ with respect to the metric induced by the standard ${\mathbb{F}}_p$ -valued Gray maps.