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Two new families of entanglement-assisted quantum MDS codes from generalized Reed–Solomon codes

Gaojun Luo, Xiwang Cao

Published Feb 9, 2019• 52 citations
Publisher

Abstract

Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC by any classical linear code. In this paper, we propose two constructions of generalized Reed–Solomon codes and calculate the dimension of their hulls. With these generalized Reed–Solomon codes, we present two new infinite families of EAQECCs, which are optimal with respect to the Singleton bound for EAQECCs. Notably, the parameters of our EAQECCs are new and flexible.

DOI
10.1007/s11128-019-2207-8

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Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC by any classical linear code. In this paper, we propose two constructions of generalized Reed–Solomon codes and calculate the dimension of their hulls. With these generalized Reed–Solomon codes, we present two new infinite families of EAQECCs, which are optimal with respect to the Singleton bound for EAQECCs. Notably, the parameters of our EAQECCs are new and flexible.
The hull of linear codes plays an important role in quantum information and coding theory. In the present paper, by investigating the Galois hulls of generalized Reed-Solomon (GRS) codes and extended GRS codes over the finite field Fq, we give several new families of MDS codes with Galois hulls of arbitrary dimensions that are not obtained before. Some of them generalize the ones in the literature [13]. As a consequence, using these MDS codes with Galois hulls of arbitrary dimensions, we construct nine new families of MDS entanglement-assisted quantum error-correcting codes (EAQECCs) with flexible parameters.
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The hull of linear codes has promising utilization in coding theory and quantum coding theory. In this paper, we study the hull of generalized Reed-Solomon codes and extended generalized Reed-Solomon codes over finite fields with respect to the Euclidean inner product. Several infinite families of MDS codes with hulls of arbitrary dimensions are presented. As an application, using these MDS codes with hulls of arbitrary dimensions, we construct several new infinite families of entanglement-assisted quantum error-correcting codes with flexible parameters.
The hull of linear codes have promising utilization in coding theory and quantum coding theory. In this paper, we study the hull of generalized Reed-Solomon codes and extended generalized Reed-Solomon codes over finite fields with respect to the Euclidean inner product. Several infinite families of MDS codes with arbitrary dimensional hull are presented. As an application, using these MDS codes with arbitrary dimensional hull, we construct several new infinite families of entanglement-assisted quantum error-correcting codes with flexible parameters.
The Hermitian hull of linear codes plays an important role in coding theory and quantum coding theory. In this paper, we first construct some infinite classes of MDS codes over finite field by considering generalized Reed-Solomon codes or extended generalized Reed- Solomon codes and determine their Hermitian hulls. The results indicate that these codes constructed have Hermitian hulls of (almost) arbitrary dimensions. Furthermore, based on these MDS codes constructed with Hermitian hulls of arbitrary dimensions, we obtain several infinite classes of entanglement-assisted quantum error-correcting (EAQEC) MDS codes whose maximally entangled states c can take almost all values. Moreover, most of the EAQEC MDS codes constructed are new in the sense that their parameters are different from all the previously known ones.
Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard quantum error-correcting codes, which can only be constructed from dual-containing codes. However, the parameter of ebits ccc is usually calculated by computer search. In this paper, we construct four classes of MDS entanglement-assisted quantum error-correcting codes (MDS EAQECCs) based on kkk-Galois LCD MDS codes for some certain code lengths, where the parameter of ebits ccc can be easily generated algebraically and not by computational search. Moreover, the constructed four classes of EAQECCs are also maximal-entanglement EAQECCs.