Galois LCD Codes Over Fq + uFq + vFq + uvFq

Abstract

In [14], Wu and Shi studied l-Galois LCD codes over finite chain ring R = F q + uF q , where u 2 = 0 and q = p e for some prime p and positive integer e. In this work, we extend the results to the finite non chain ring R = F q + uF q + vF q + uvF q , where u 2 = u, v 2 = v and uv = vu. We define a correspondence between l-Galois dual of linear codes over R and l-Galois dual of its component codes over F q . Further, we construct Euclidean LCD and l-Galois LCD codes from linear code over R. This consequently leads us to prove that any linear code over R is equivalent to Euclidean (q > 3) and l-Galois LCD (0 < l < e, and p e−l + 1 | p e − 1) code over R. Finally, we investigate MDS codes over R.