New Quantum Codes and (<italic>&#x03B8;</italic>, <italic>&#x03B4;</italic>, <italic>&#x03B2;</italic>)-Cyclic Codes

Abstract

For a prime p and a positive integer e, let $\mathbb {F}_{p^{e}}$ be the finite field of order $p^{e}$ and $\mathfrak {R}_{\ell } $ be a non-chain ring given by $\mathfrak {R}_{\ell } :=\mathbb {F}_{p^{e}}[v]/\langle v^{\ell } -1\rangle,~\ell \gt 1$ . This study presents the $(\theta _{1},\delta _{1},\beta _{1})$ -cyclic codes over $\mathfrak {R}_{\ell } $ and $(\theta _{i},\delta _{i},\beta _{i})$ -cyclic codes over $\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell } $ for $i=0,1$ . Further, we study the application of these codes in the construction of quantum codes. Towards this, we define a Gray map to find images of the elements of $\mathfrak {R}_{\ell } $ and $\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell } $ over the copies of $\mathbb {F}_{p^{e}}$ , and then establish the dual-containing conditions for these codes. Finally, using the CSS construction and a propagation rule, we find many non-binary quantum and classical codes with better parameters.

Keywords

• (θ, δ, β)-cyclic code
• quantum code
• non-chain ring
• additive code