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Entropy

On Optimal and Quantum Code Construction from Cyclic Codes over FqPQ with Applications

Shakir Ali, Amal S. Alali, Pushpendra Sharma, Kok Bin Wong, Elif Segah Öztaş, Mohammad Jeelani

Published Aug 2, 2023• 4 citations
Publisher

Abstract

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.

DOI
10.3390/e25081161

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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.
On the Structure of Cyclic Codes Over FqRS\mathbb{F}_{q}{RS}Fq​RS and Applications in Quantum and LCD Codes Constructions
Let p be an odd prime, q = pm, R = Fq+ uFqwith u2= 1, and S = Fq+ uFq+ vFq+ uvFqwith u2= 1, v2= 1, uv = vu. In this paper, FqRS-cyclic codes over FqRS are studied. As an application, we present a construction of quantum error-correcting codes (QECCs) from the FqRS-cyclic codes over FqRS, which provides new QECCs. We also consider linear complementary dual (LCD) codes from the FqRS-cyclic codes over FqRS. Among others, we construct a Gray map over FqRS and discuss the Gray images of FqRS-cyclic codes over Fq.
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.
Quantum codes constructed from cyclic codes over the ring Fq+vFq+v2Fq+v3Fq+v4Fq\mathbb{F}_q+v\mathbb{F}_q+v^2\mathbb{F}_q+v^3\mathbb{F}_q+v^4\mathbb{F}_qFq​+vFq​+v2Fq​+v3Fq​+v4Fq​
In this article, we investigate properties of cyclic codes over a finite non-chain ring Fq+vFq+v2Fq+v3Fq+v4Fq,\mathbb{F}_q+v\mathbb{F}_q+v^2\mathbb{F}_q+v^3\mathbb{F}_q+v^4\mathbb{F}_q,Fq​+vFq​+v2Fq​+v3Fq​+v4Fq​, where q=pr,q=p^r,q=pr, rrr is a positive integer, ppp is an odd prime, 4∣(p−1),4 \mid (p-1),4∣(p−1), and v5=v.v^5=v.v5=v. As an application, we construct several quantum error correcting codes over the finite field Fq.\mathbb{F}_q.Fq​.