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AIP conference proceedings

Quantum error correction through constacyclic codes over mixed alphabet

Mohd Azmi, Mohammad Fareed Ahmad, Nadeem ur Rehman, Arifa Naz

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Let q=pr, where p is an odd prime and r be a positive integer, 𝒮=𝔽q[u]〈u2−1〉 and 𝒯=𝔽q[u,v]〈u2−1,v2−1,uv=vu〉 and Ω=(ω1,ω2,ω3)∈𝔽q𝒮𝒯⁠. The algebraic structure of constacyclic codes over 𝔽q𝒮𝒯−Ω–constacyclic code of block length (l, m, n) is examined in this paper. The construction of quantum error-correcting codes (QECCs) from constacyclic codes over 𝔽q𝒮𝒯 is also covered in this paper and we derive some new QECCs.

DOI
10.1063/5.0296776

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Let q=pr, where p is an odd prime and r be a positive integer, 𝒮=𝔽q[u]〈u2−1〉 and 𝒯=𝔽q[u,v]〈u2−1,v2−1,uv=vu〉 and Ω=(ω1,ω2,ω3)∈𝔽q𝒮𝒯⁠. The algebraic structure of constacyclic codes over 𝔽q𝒮𝒯−Ω–constacyclic code of block length (l, m, n) is examined in this paper. The construction of quantum error-correcting codes (QECCs) from constacyclic codes over 𝔽q𝒮𝒯 is also covered in this paper and we derive some new QECCs.
Let [Formula: see text], where p is an odd prime and r is a positive integer. Consider a ring [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text] with [Formula: see text]. In this paper, we examine the algebraic structure of constacyclic codes over the ring [Formula: see text] of block length [Formula: see text]. Further, a construction of quantum error-correcting codes (QECCs) from constacyclic codes over [Formula: see text] is also given. Moreover, we derive a number of new QECCs.
The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of FqPQ, where P=Fq[v]⟨v3−α22v⟩ and Q=Fq[u,v]⟨u2−α12,v3−α22v⟩ are nonchain finite rings and αi is in Fq/{0} for i∈{1,2}, where q=pm with m≥1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes.
In this article, we construct some MDS quantum error-correcting codes (QECCs) from classes of constacyclic codes over Rs= Fp+ u1Fp+ ··· + usFp, ui2= ui, uiuj= ujui= 0, for odd prime p and i, j = 1, 2,⋯, s, i ≠ j. Many QECCs with improved parameters than the existing ones in some of the earlier papers are provided. We present a set of idempotent generators of the ring Rs, and using that we define linear codes, determine all units, and study constacyclic codes over this ring. Among others, we study dual containing constacyclic codes over Rs and construct (non-binary) QECCs. An algorithm to construct QECCs from dual containing constacyclic codes over Rs is obtained that can provide many quantum codes.
This paper focuses on studying the properties of constacyclic codes, quantum error-correcting codes. The code is studied over a specific mathematical structure called the ring S\mathfrak{S}S, which is defined as S=Iq[u,v]/⟨u2−α2, v2−α2, uv−vu⟩\mathfrak{S}=\mathfrak{I}_q[\mathfrak{u},\mathfrak{v}]/\langle \mathfrak{u}^2-\alpha^2,~ \mathfrak{v}^2-\alpha^2,~\mathfrak{u}\mathfrak{v}-\mathfrak{v}\mathfrak{u} \rangleS=Iq​[u,v]/⟨u2−α2, v2−α2, uv−vu⟩, where Iq\mathfrak{I}_qIq​ is a finite field of qqq elements, α\alphaα be the nonzero elements of the field Iq\mathfrak{I}_qIq​ and qqq is a power of an odd prime ppp such that q=pm, for m≥1q=p^m, ~\textup{for}~ m \ge 1q=pm, for m≥1. The paper also introduces a Gray map and use it to decompose constacyclic codes over the ring S\mathfrak{S}S into a direct sum of constacyclic codes over Iq\mathfrak{I}_qIq​. We construct new and better quantum error-correcting codes over the ring S\mathfrak{S}S (cf.; Table 1 and Table 2). Moreover, we also obtain best known linear codes as well as best dimension linear codes (cf.; Table 4).
Let R = Fq+ uFq+ vFq+ uvFq, with u2= u, v2= v, uv = vu, where q = pmfor a positive integer m and an odd prime p. We study the algebraic structure of FqR-cyclic codes of block length (r, s). These codes can be viewed as R[x]-submodules of Fq[x]/(xr- 1) × R[x]/(xs- 1). For this family of codes we discuss the generator polynomials and minimal generating sets. We study the algebraic structure of separable codes. Further, we discuss the duality of this family of codes and determine their generator polynomials. We obtain several optimal and near-optimal codes from this study. As applications, we discuss a construction of quantum error-correcting codes (QECCs) from FqR-cyclic codes and construct some good QECCs.