Cyclic codes over a non-chain ring R e, q and their application to LCD codes.

Abstract Let F q be a finite field of order q, a prime power integer, such that q = e t + 1 where t ≥ 1 , e ≥ 2 are integers. In this paper, we study cyclic codes of length n over a non-chain ring R e , q = F q [ u ] / 〈 u e − 1 〉 . We define a Gray map φ and obtain many maximum-distance-separable (MDS) and optimal F q -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 F q . 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 R e , q .