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Axioms

On the Construction of Quantum and LCD Codes from Cyclic Codes over the Finite Commutative Rings

Shakir Ali, Amal S. Alali, Mohammad Jeelani, Muhammet Kurulay, Elif Segah Öztaş, Pushpendra Sharma

Published Apr 10, 2023• 7 citations
Publisher

Abstract

Let Fq be a field of order q, where q is a power of an odd prime p, and α and β are two non-zero elements of Fq. The primary goal of this article is to study the structural properties of cyclic codes over a finite ring R=Fq[u1,u2]/⟨u12−α2,u22−β2,u1u2−u2u1⟩. We decompose the ring R by using orthogonal idempotents Δ1,Δ2,Δ3, and Δ4 as R=Δ1R⊕Δ2R⊕Δ3R⊕Δ4R, and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes.

DOI
10.3390/axioms12040367

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Let Fq be a field of order q, where q is a power of an odd prime p, and α and β are two non-zero elements of Fq. The primary goal of this article is to study the structural properties of cyclic codes over a finite ring R=Fq[u1,u2]/⟨u12−α2,u22−β2,u1u2−u2u1⟩. We decompose the ring R by using orthogonal idempotents Δ1,Δ2,Δ3, and Δ4 as R=Δ1R⊕Δ2R⊕Δ3R⊕Δ4R, and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD 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.
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.
This paper primarily investigates the structural properties of constacyclic codes over the ring , defined as where i, j = 1, 2, 3, ..., k, and k, m are positive integers. Here, denotes a finite field of order pm with characteristic p, an odd prime. Furthermore, we determine the necessary and sufficient conditions for the duals of constacyclic codes to exist. These findings make it easier to create new Quantum Error-Correcting (QEC) codes throughout the ring (i.e., when k = 1), as well as optimal linear codes that make use of the Gray images of constacyclic codes. Additionally, Table 4 presents several Linear Complementary Dual (LCD) codes obtained using the Gray map.
For an integer m ≥ 1, we study cyclic codes of length l over a commutative non-chain ring F2m + uF2m , where u2 = u . With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length l with respect to their generator polynomials and prove that F2m − image of an LCD code of length l is an LCD code of length 2l . Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m .
Let k,m be positive integers and F2m be a finite field of order 2m of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring Sk=F2m[v1,v2,…,vk]⟨vi2−αivi,vivj−vjvi⟩, for i,j=1,2,3,…,k, where αi is the non-zero element of F2m. As an application, we obtain better quantum error correcting codes over the ring S1 (for k=1). Moreover, we acquire optimal linear codes with the help of the Gray image of cyclic codes. Finally, we present methods for reversible DNA codes.