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On the hull-variation problem of Equivalent Linear Codes

IEEE TRANSACTIONS ON INFORMATION THEORY 2023年第5期69卷 2911-2922页

作者： Chen, Hao Jinan Univ Coll Informat Sci & Technol Guangzhou 510632 Guangdong Peoples R China Jinan Univ Coll Cyber Secur Guangzhou 510632 Guangdong Peoples R China

The intersection C boolean AND C perpendicular to (C boolean AND C-perpendicular to h) of a linear code C and its Euclidean dual C perpendicular to (Hermitian dual C-perpendicular to h) is called the Euclidean (Hermitian) hull of this code. It is natural to consider the hull-variation problem when a linear code C is transformed to an equivalent code v center dot C. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. We prove that for a nonnegative integer h satisfying 0 <= h <= n - 1, a linear [2n, n] q self-dual code is equivalent to a linear h-dimension hull code. On the opposite direction we prove that a linear LCD code over F2s satisfying d >= 2 and d(perpendicular to)>= 2 is equivalent to a linear one-dimension hull code under a weak condition. Several new families of LCD negacyclic codes and LCD BCH codes over F3 are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new entanglement-assisted quantum error-correction (EAQEC) codes including MDS and almost MDS EAQEC codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.

关键词： hull-variation problem maximal hull dimension self-dual code LCD code linear h-dimension hull code EAQEC code

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On symplectic hulls of linear codes and related applications

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2024年第3期70卷 2603-2622页

作者： Li, Yang Zhu, Shixin Hefei Univ Technol Sch Math Hefei 230601 Anhui Peoples R China

This paper investigates symplectic hulls of linear codes. We use a different view to obtain more structural properties of generator matrices with respect to the symplectic inner product. As an outgrowth, generalized formulas for calculating dimensions of symplectic hulls are derived, which extend some known results in the literature. We then study the symplectic hull-variation problem and prove that a monomially equivalent linear code with a smaller dimensional symplectic hull can always be explicitly derived from a given q-ary symplectic self-dual code with a standard generator matrix for q >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\ge 3$$\end{document}. As an application, we present an improved propagation rule for constructing entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain some new and record-breaking binary EAQECCs.

关键词： Symplectic hull Monomial equivalence Permutation equivalence hull-variation problem Entanglement-assisted quantum error-correcting code

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