Abstract
For an odd prime p and a positive integer r, let \(q=p^r\). The objective of this article is to study cyclic codes over the ring \(\mathcal {S}=\mathbb {F}_q[\textsf{u}, \textsf{v},\textsf{w}]/\langle \textsf{u}^3-\textsf{u},\textsf{v}^2-\textsf{v}, \textsf{w}^2-\textsf{w}, \textsf{u}\textsf{v}, \textsf{v}\textsf{u}, \textsf{u}\textsf{w}, \textsf{w}\textsf{u}, \textsf{v}\textsf{w}-\textsf{w}\textsf{v} \rangle \) and to construct new and better quantum and LCD codes from them. We give the structure of cyclic codes over the ring \(\mathcal {S}\) and obtain dual-containing codes over \(\mathbb {F}_q\) as the Gray images of dual-containing cyclic codes over \(\mathcal {S}\). Using these dual-containing codes, we obtain quantum codes and determine their parameters using the decomposition of cyclic codes over the ring \(\mathcal {S}\). We provide many new and better-than-existing quantum codes. We also give a method to obtain linear complementary dual (LCD) codes over \(\mathcal {S}\) using the decomposition of cyclic codes over the ring \(\mathcal {S}\). We obtain some optimal and best-known linear codes as Gray images of LCD codes over \(\mathcal {S}\).
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Rai, P., Singh, B. & Gupta, A.J. On quantum and LCD codes from the cyclic codes over the ring \(\mathbb {F}_q[\textsf{u}, \textsf{v},\textsf{w}]/\langle \textsf{u}^3-\textsf{u},\textsf{v}^2-\textsf{v}, \textsf{w}^2-\textsf{w}, \textsf{u}\textsf{v}, \textsf{v}\textsf{u}, \textsf{u}\textsf{w}, \textsf{w}\textsf{u}, \textsf{v}\textsf{w}-\textsf{w}\textsf{v} \rangle \). J. Appl. Math. Comput. 70, 1241–1262 (2024). https://doi.org/10.1007/s12190-024-02002-w
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DOI: https://doi.org/10.1007/s12190-024-02002-w