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Advances in Mathematics of Communications

Construction of linear codes with various Hermitian hull dimensions and related EAQECCs

Ruowen Liu, Shitao Li, Minjia Shi

Published Mar 29, 2024• 4 citations
Publisher

Abstract

The hull of a linear code is the intersection of the code and its dual code, which is effective for determining parameters of entanglement-assisted quantum error-correcting codes (EAQECCs). There are few constructions of linear codes with various Hermitian hull dimensions, aside from Hermitian LCD and self-orthogonal codes. The object of this paper is to introduce a building-up construction for constructing linear $ [n+2, k+1] $ codes with $ \ell $ or $ (\ell+1) $-dimensional Hermitian hull from a given linear $ [n, k] $ code with $ \ell $-dimension Hermitian hull of a smaller length. This construction includes the converse of the famous shortening technique as a special case. Using this method, we construct optimal quaternary linear codes of lengths up to 13 with Hermitian hull dimensions 2-5. As an application, we construct many EAQECCs, which improve the parameters of EAQECCs of Grassl's code table.

DOI
10.3934/amc.2024013

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The hull of a linear code is the intersection of the code and its dual code, which is effective for determining parameters of entanglement-assisted quantum error-correcting codes (EAQECCs). There are few constructions of linear codes with various Hermitian hull dimensions, aside from Hermitian LCD and self-orthogonal codes. The object of this paper is to introduce a building-up construction for constructing linear [n+2,k+1][n+2, k+1][n+2,k+1] codes with ℓ\ellℓ or (ℓ+1)(\ell+1)(ℓ+1)-dimensional Hermitian hull from a given linear [n,k][n, k][n,k] code with ℓ\ellℓ-dimension Hermitian hull of a smaller length. This construction includes the converse of the famous shortening technique as a special case. Using this method, we construct optimal quaternary linear codes of lengths up to 13 with Hermitian hull dimensions 2-5. As an application, we construct many EAQECCs, which improve the parameters of EAQECCs of Grassl's code table.
Linear codes with small hulls over finite fields have been extensively studied due to their practical applications in computational complexity and information protection. In this paper, we develop a general method to determine the exact value of D4H(n,k,1)D_4^H(n,k,1)D4H​(n,k,1) for n≤12n\leq 12n≤12 or k∈{1,2,3,n−1,n−2,n−3}k\in \{1,2,3,n-1,n-2,n-3\}k∈{1,2,3,n−1,n−2,n−3}, where D4H(n,k,1)D_4^H(n,k,1)D4H​(n,k,1) denotes the largest minimum distance among all quaternary linear [n,k][n,k][n,k] codes with one-dimensional Hermitian hull. As a consequence, we solve a conjecture proposed by Mankean and Jitman on the largest minimum distance of a quaternary linear code with one-dimensional Hermitian hull. As an application, we construct some binary entanglement-assisted quantum error-correcting codes (EAQECCs) from quaternary linear codes with one-dimensional Hermitian hull. Some of these EAQECCs are optimal codes, and some of them are better than previously known ones.
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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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