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Journal of Applied Mathematics and Computing
Erschienen in: Journal of Applied Mathematics and Computing 4/2022

12.09.2021 | Original Research

A family of constacyclic codes over a class of non-chain rings \({\mathcal {A}}_{q,r}\) and new quantum codes

verfasst von: Habibul Islam, Shikha Patel, Om Prakash, Patrick Solé

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 4/2022

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Abstract

For a prime p, let \({\mathbb {F}}_{q^2}\) be the finite field of \(q^2\) elements where \(q=p^m\) and \(m\ge 1\) is an integer. In this paper, we study constacyclic and skew constacyclic codes of length n over a class of finite commutative non-chain rings \({\mathcal {A}}_{q,r}={\mathbb {F}}_{q^2}[u_1,u_2,\dots ,u_r]/\langle u^2_i-\gamma _i u_i,u_iu_j=u_ju_i=0\rangle \) where \(1\le i\ne j \le r\), \(\gamma _i \in {\mathbb {F}}_{q^2}^*\), and \(r\ge 1\) is an integer. For a unit \(\varLambda =\sum \nolimits \nolimits _{i=0}^r\kappa _i\varLambda _i\) in the ring \({\mathcal {A}}_{q,r},\) we show that a (skew) \(\varLambda \)-constacyclic code of length n is a direct sum of (skew) \(\varLambda _i\)-constacyclic codes of length n over \({\mathbb {F}}_{q^2}\). Also, we derive the necessary and sufficient conditions for such codes (constacyclic and skew constacyclic) to contain their Hermitian duals. By applying the Hermitian construction on the Gray images of dual containing codes, many MDS and new quantum codes with parameters better than the best-known codes are constructed.
Vorheriger Artikel ADI Galerkin finite element scheme for the two-dimensional semilinear partial intergro-differential equation with a weakly singular kernel
Nächster Artikel Sufficient conditions for the existence of oscillatory solutions to nonlinear second order differential equations

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Metadaten
Titel
A family of constacyclic codes over a class of non-chain rings and new quantum codes
verfasst von
Habibul Islam
Shikha Patel
Om Prakash
Patrick Solé
Publikationsdatum
12.09.2021
Verlag
Springer Berlin Heidelberg
Erschienen in
Journal of Applied Mathematics and Computing / Ausgabe 4/2022
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-021-01623-9

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