Constructions and Enumerations of Self-Dual and LCD Double Circulant Codes over a Local Ring

The construction of self-dual and linear complementary dual (LCD) codes over finite rings, particularly over semi-local and local structures, is an active area of research due to their algebraic richness and applications in communications and cryptography. In this paper, we investigate double circulant and double negacirculant codes over the local ring Rq,u,v=Fq+uFq+vFq,u2=v2=uv=vu=0, where q=pm is an odd prime power. Unlike the semi-local case, where decomposition via non-trivial idempotents simplifies analysis, the local structure of Rq,u,v (with only trivial idempotents) makes enumeration and classification significantly more challenging. We first establish necessary and sufficient conditions for such codes to be self-dual or LCD; we then count the solutions to key equations over Fq, including abq+baq=0, to enable their enumeration. We further show that Gray images preserve these properties, leading to good self-dual and LCD codes over Fq, and present optimal examples over F7. Our results extend double circulant constructions to a new algebraic setting, providing both theoretical advancements and practically relevant code designs.