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New Quantum Codes and (θ, δ, β)-Cyclic Codes

Ashutosh Singh, Priyanka Sharma, Om Prakash

Published Jan 1, 2024• 3 citations
Publisher

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.

DOI
10.1109/access.2024.3419099

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