Cyclic codes over a non-chain ring Re,q and their application to LCD codes

Abstract

Let Fq be a finite field of order q, a prime power integer, such that q=et+1 where t≥1,e≥2 are integers. In this paper, we study cyclic codes of length n over a non-chain ring Re,q=Fq[u]/〈ue−1〉. We define a Gray map φ and obtain many maximum-distance-separable (MDS) and optimal Fq-linear codes from the Gray images of cyclic codes. Under certain conditions we determine linear complementary dual (LCD) codes of length n when gcd⁡(n,q)≠1 and gcd⁡(n,q)=1, respectively. It is proved that a cyclic code C of length n is an LCD code if and only if its Gray image φ(C) is an LCD code of length en over Fq. Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over Re,q.