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

For 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.