A NOTE ON k-GALOIS LCD CODES OVER THE RING

Abstract

We study the k-Galois linear complementary dual (LCD) codes over the finite chain ring $R=\mathbb {F}_q+u\mathbb {F}_q$ with $u^2=0$ , where $q=p^e$ and p is a prime number. We give a sufficient condition on the generator matrix for the existence of k-Galois LCD codes over R. Finally, we show that a linear code over R (for $k=0, q> 3$ ) is equivalent to a Euclidean LCD code, and a linear code over R (for $0<k<e$ , $(p^{e-k}+1)\mid (p^e-1)$ and ${(p^e-1)}/{(p^{e-k}+1)}>1$ ) is equivalent to a k-Galois LCD code.

Galois LCD Codes Over Fq + uFq + vFq + uvFq

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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 .

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“…Several authors investigated skew cyclic codes, constacyclic codes and quantum error correcting codes over ring R [16,9,1]. The study of l-Galois LCD codes over finite chain ring F q + uF q was done in [14]. Authors proved that for any linear code over F q + uF q there exists equivalent Euclidean and l-Galois LCD code.…”

“…The main idea of this article is to deal with this problem. Taking inspiration from [14], we characterize l-Galois linear codes over finite non chain ring R = F q + uF q + vF q + uvF q . The outline of this article is arranged as follows.…”

Galois LCD Codes Over

“…Prakash et al [14] enumerated self-dual and LCD double circulant codes over a class of finite commutative nonchain rings and investigated the algebraic structure of 1-generator quasi-cyclic (QC) codes over for . The -Galois LCD codes over the finite chain ring are studied in [15], showing that for any linear code over , there exist equivalent Euclidean and -Galois LCD codes. Taking inspiration from [15], we consider -Galois linear codes over the finite nonchain ring, and characterise -Galois LCD codes over this ring.…”

“…The -Galois LCD codes over the finite chain ring are studied in [15], showing that for any linear code over , there exist equivalent Euclidean and -Galois LCD codes. Taking inspiration from [15], we consider -Galois linear codes over the finite nonchain ring, and characterise -Galois LCD codes over this ring.…”