arXiv:2002.12892

Galois hulls of MDS codes and their quantum error correction

Meng Cao

Published Feb 28, 2020• 3 citations

Abstract

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.

DOI: 10.48550/arxiv.2002.12892

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arXiv:2002.12892

Galois hulls of MDS codes and their quantum error correction

Meng Cao

Published Feb 28, 2020• 3 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2002.12892

Related papers

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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

-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

-ary MDS codes in this paper enables us to obtain new

-ary EAQECCs with different kinds of length sets via different

\ell $, where

2\ell \mid e

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes over

\mathbb {F}_{q^{m}}

are generalized Reed-Solomon codes (GRS codes) when

m=1

. In this paper, we give a necessary and sufficient condition for the Galois dual codes of Goppa codes when

m=1

to be Goppa codes with the same locator sets. Furthermore, we show that the Galois hulls of the above codes are still the Goppa codes and determine their Goppa polynomials and dimensions. In particular, we characterize Galois linear complementary dual (LCD), Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such codes. Moreover, we apply these results to EAQECCs.