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IEEE Transactions on Information Theory

MDS Codes With Galois Hulls of Arbitrary Dimensions and the Related Entanglement-Assisted Quantum Error Correction

Meng Cao

Published Oct 6, 2021• 49 citations
Publisher

Abstract

Let$q=p^{e}$be a prime power and$\ell $be an integer with$0\leq \ell \leq e-1$. The$\ell $-Galois hull of classical linear codes is a generalization of the Euclidean hull and Hermitian hull. We provide a necessary and sufficient condition under which a codeword of a GRS code or an extended GRS code belongs to its$\ell $-Galois dual code, generalizing both the Euclidean case and Hermitian case in the literature. By using four different tools: 1) the norm mapping from$\mathbb {F}_{q}^{\ast }$to$\mathbb {F}_{p^{\ell }}^{\ast }$; 2) the direct product of two cyclic subgroups; 3) the coset decomposition of a cyclic group; 4) an additive subgroup of$\mathbb {F}_{q}$and its cosets, we construct eleven families of$q$-ary MDS codes with$\ell $-Galois hulls of arbitrary dimensions, and give the related eleven families of$[[n,k,d;c]]_{q}$entanglement-assisted quantum error-correcting codes (EAQECCs) with relatively large minimum distance in the sense that$2d=n-k+2+c$. We show that developing the theory on$\ell $-Galois hulls of$q$-ary MDS codes in this paper enables us to obtain new$q$-ary EAQECCs with different kinds of length sets via different$\ell $, where$2\ell \mid e$.

DOI
10.1109/tit.2021.3117562

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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.
In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length n > q+1 are also obtained.
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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.