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IEEE Access

A Study of FqR-Cyclic Codes and Their Applications in Constructing Quantum Codes

Hai Quang Dinh, Sachin Pathak, Tushar Bag, Ashish Kumar Upadhyay, Warattaya Chinnakum

Published Jan 1, 2020• 15 citations
Publisher

Abstract

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.

DOI
10.1109/access.2020.3032078

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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.
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.
QUANTUM CODES FROM CYCLIC CODES OVER Fq+uFq+vFq\mathbb{F}_{q}+u \mathbb{F}_{q}+v \mathbb{F}_{q}Fq​+uFq​+vFq​
In this paper, we investigate the method of constructing quantum codes from cyclic codes over the ring R=Fq+uFq+vFqR= \mathbb{F}_{q}+u \mathbb{F}_{q}+v \mathbb{F}_{q}R=Fq​+uFq​+vFq​ (u2=u, v2=v, uv=vu=0). By means of the generator polynomials and some decomposition of cyclic codes and their dual over R, we give a necessary and sufficient condition of the cyclic codes over R to be construct the quantum codes. Also as an application, we obtain new non binary quantum codes from those classes of cyclic codes.
Let R= Fq+uFq+vFq+uvFq be a commutative ring with u2=u, v2=v, uv=vu, where q is a power of an odd prime. Ashraf and Mohammad constructed some new quantum codes from cyclic codes. Under this background, another Gray map from R to Fq4 is given. This map can be naturally extended to Rn. The problem on the ring turns to the field by this isomorphic map now. Therefore, This mapping is obviously a weight-preserving and distance-preserving map. The results show that the codes after mapping are self-orthogonal codes over Fq if they are self-orthogonal codes over R. Some computational examples show that some better non-binary quantum codes can be obtained under this Gray map. We discuss the structure of linear codes. On this basis, the structure of the generating matrix of linear codes is obtained. The structure of their dual codes is also obtained. The CSS construction guarantees the existence of quantum codes. Finally, with the help of the CSS construction, we get some good quantum codes. By comparison, our quantum codes have better parameters.
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.