Self-Dual Double Circulant, Self-Dual Double Negacirculant and LCD Double Negacirculant Codes Over the Ring F_q[u,v]/<u² - u, v²-v, uv-vu>

Abstract

In this paper, we investigate self-dual double circulant, and self-dual and linear complementary dual (LCD) double negacirculant codes over a finite ring R = F q + uF q + vF q + uvF q , where u 2 = u, v 2 = v, uv = vu and q = p m . We study the algebraic structure of double circulant codes over R. We provide necessary and sufficient conditions for a double circulant code to be a self-dual code. We give a formula to get the total number of self-dual double circulant codes over the ring R. We compute distance bounds for self-dual double circulant codes over R. In addition, by using a Gray map, we show that the families of selfdual double circulant codes under the Gray map are asymptotically good. Moreover, the algebraic structure of double negacirculant codes and necessary and sufficient conditions for a double negacirculant code to be a self-dual code and to be an LCD code are also given. We determine the total number of self-dual and LCD double negacirculant codes over R.INDEX TERMS Double circulant codes; double negacirculant codes; self-dual codes; LCD codes; Gray map; Artin conjecture.