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 $R = \mathbb F_{q} + u \mathbb F_{q} + v \mathbb F_{q} + uv\mathbb F_{q}$ , 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 self-dual 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 $R$ .

Keywords

• Double circulant codes
• double negacirculant codes
• self-dual codes
• LCD codes
• Gray map
• Artin conjecture