On self-dual and LCD double circulant and double negacirculant codes over Fq+uFq$\mathbb {F}_{q}+u\mathbb {F}_{q}$

Abstract

Double circulant codes of length 2n over the non-local 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 for n →∞. The parameters of examples of modest lengths are computed. Several such codes are optimal.