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arXiv:2210.14549 · Computational and Applied Mathematics

Binary optimal linear codes with various hull dimensions and entanglement-assisted QECCs

Jon-Lark Kim

Published Mar 16, 2023• 12 citations
arXivHTMLPDF
DOI
10.1007/s40314-023-02268-z

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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.