Cyclic and LCD Codes over a Finite Commutative Semi-local Ring

Abstract

AbstractFor an odd prime p, we obtain algebraic structure of cyclic codes of length n over a finite commutative non-chain semi-local ring \(\mathfrak {R}=\mathbb {F}_{p}[u,v,w]/\langle u^{2}-u,v^{2}-1,w^2-1,uv-vu,vw-wv,wu-uw\rangle \). These codes of length n can be viewed as principal ideals of the quotient ring \(\mathfrak {R}[x]/\langle x^n-1\rangle \). Here, a Gray map is defined to obtain p-ary quasi-cyclic codes of length 8n with index 8 as \(\mathbb {F}_p\)-images of cyclic codes of length n over \(\mathfrak {R}\). Also, we present necessary and sufficient conditions for a cyclic code to be an LCD (linear complementary dual) code over \(\mathfrak {R}\). Moreover, it is shown that the Gray image of an LCD code of length n over \(\mathfrak {R}\) is an LCD code of length 8n over \(\mathbb {F}_{p}\). Finally, a few non-trivial examples are given in support of our derived results.KeywordsCyclic codeNon-chain ringSemi-local ringGray mapLCD code