Construction of LCD and new quantum codes from cyclic codes over a finite non-chain ring

Abstract

For an odd prime p and q = pr, this paper deals with LCD codes obtained from cyclic codes of length n over a finite commutative non-chain ring $\\mathcal {R}=\\mathbb {F}_{q}[u,v]/\\langle u^{2}-\\alpha u,v^{2}-1, uv-vu\\rangle $\n where α is a non-zero element in $\\mathbb {F}_{q}$\n . Initially, we impose certain conditions on the generator polynomials of cyclic codes when $\\gcd (n,p)=1$\n and $\\gcd (n,p)\\neq 1$\n , respectively so that these codes become LCD. Then, by defining a Gray map ψ, we show that the Gray image of an LCD code of length n over $\\mathcal {R}$\n is an LCD code of length 4n over $\\mathbb {F}_{q}$\n . In this way, we obtain many optimal and best-known linear codes (BKLC) from the Gray images of both cyclic and LCD codes over $\\mathcal {R}$\n . Eventually, by applying the CSS construction on cyclic codes over $\\mathcal {R}$\n that contain their Euclidean duals, we determine many superior quantum codes compared to the existing codes in the recent references.