Journal of Applied Mathematics and Computing | 2021
Self-dual and LCD double circulant and double negacirculant codes over $${\\mathbb {F}}_q+u{\\mathbb {F}}_q+v{\\mathbb {F}}_q$$
Abstract
Let q be an odd prime power, and denote by $${\\mathbb {F}}_q$$\n the finite field with q elements. In this paper, we consider the ring $$R={\\mathbb {F}}_q+u{\\mathbb {F}}_q+v{\\mathbb {F}}_q$$\n , where $$u^2=u, v^2=v,uv=vu=0$$\n and study double circulant and double negacirculant codes over this ring. We first obtain the necessary and sufficient conditions for a double circulant code to be self-dual (resp. LCD). Then we enumerate self-dual and LCD double circulant and double negacirculant codes over R. Last but not the least, we show that the family of Gray images of self-dual and LCD double circulant codes over R are good. Several numerical examples of self-dual and LCD codes over $${\\mathbb {F}}_5$$\n as the Gray images of these codes over R are given in short lengths.