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IEEE International Students' Conference on Electrical, Electronics and Computer Science (SCEECS)

Linear l-intersection Pairs of Monomial and Decreasing Monomial Cartesian Codes with Application to EAQEC Codes *

Md Ajaharul Hossain, Ramakrishna Bandi

Published Feb 18, 2023• 0 citations
Publisher

Abstract

Linear l-intersection pair of codes is the general-ization of LCP of codes and hull of codes. The$\ell$represents the dimension of the intersection of a given pair of codes over a finite field$\mathbb{F}_{q}$. The l-intersection is useful in obtaining some good Entanglement assisted quantum error-correcting (EAQEC) codes and for some cases MDS EAQEC codes. In this paper, the l-intersection is studied on Monomial Cartesian (MC) and Decreasing Monomial Cartesian (DMC) codes, and their char- acterizations are given. As an application, some good parameter EAQEC codes are obtained and presented in a tabular form.

DOI
10.1109/sceecs57921.2023.10063066

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Linear l-intersection pair of codes is the general-ization of LCP of codes and hull of codes. Theℓ\ellℓrepresents the dimension of the intersection of a given pair of codes over a finite fieldFq\mathbb{F}_{q}Fq​. The l-intersection is useful in obtaining some good Entanglement assisted quantum error-correcting (EAQEC) codes and for some cases MDS EAQEC codes. In this paper, the l-intersection is studied on Monomial Cartesian (MC) and Decreasing Monomial Cartesian (DMC) codes, and their char- acterizations are given. As an application, some good parameter EAQEC codes are obtained and presented in a tabular form.
Generalizing the linear complementary duals, the linear complementary pairs and the hull of codes, we introduce the concept of ℓ\ellℓ-dimension linear intersection pairs (ℓ\ellℓ-DLIPs) of codes over a finite commutative ring (R)(R)(R), for some positive integer ℓ\ellℓ. In this paper, we study ℓ\ellℓ-DLIP of codes over RRR in a very general setting by a uniform method. Besides, we provide a necessary and sufficient condition for the existence of a non-free (or free) ℓ\ellℓ-DLIP of codes over a finite commutative Frobenius ring. In addition, we obtain a generator set of the intersection of two constacyclic codes over a finite chain ring, which helps us to get an important characterization of ℓ\ellℓ-DLIP of constacyclic codes. Finally, the ℓ\ellℓ-DLIP of constacyclic codes over a finite chain ring are used to construct new entanglement-assisted quantum error correcting (EAQEC) codes.
Recently, Galindo et al. introduced the concept of asymmetric entanglement-assisted quantum error-correcting codes (AEAQECCs) from Calderbank-Shor-Steane (CSS) construction. In general, it's difficult to determine the required number of maximally entangled states of an AEAQECC, which is associated with the dimension of the intersection of the two corresponding linear codes. Two linear codes are said to be a linear l-intersection pair if their intersection has dimension l. In this paper, all possible linear l-intersection pairs of MDS codes are given. As an application, we give a complete characterization of pure MDS AEAQECCs for all possible parameters.
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Linear codes over finite fields with small dimensional hulls have received much attention due to their applications in cryptology and quantum computing. In this paper, we study cyclic and negacyclic codes with one-dimensional hulls. We determine precisely when cyclic and negacyclic codes over finite fields with one-dimensional hulls exist. We also introduce one-dimensional linear complementary pairs of cyclic and negacyclic codes. As an application, we obtain numerous optimal or near optimal cyclic codes with one-dimensional hulls over different fields and, by using these codes, we present new entanglement-assisted quantum error-correcting codes (EAQECCs). In particular, some of these EAQEC codes are maximal distance separable (MDS). We also obtain one-dimensional linear complementary pairs of cyclic codes, which are either optimal or near optimal.