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High-Rate quantum Low-Density Parity-Check codes assisted by Reliable Qubits

IEEE TRANSACTIONS ON INFORMATION THEORY 2015年第4期61卷 1860-1878页

作者： Fujiwara, Yuichiro Gruner, Alexander Vandendriessche, Peter CALTECH Div Phys Math & Astron Pasadena CA 91125 USA Mercedes Benz Bank AG D-70469 Stuttgart Germany Univ Ghent Dept Math B-9000 Ghent Belgium

quantum error correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error-correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum information science, where in principle any binary linear code can be turned into a quantum error-correcting code by assuming a small number of reliable quantum bits. This paper studies how best to take advantage of these latest ideas to construct desirable quantum error-correcting codes of very high information rate. Our methods exploit structured high-rate low-density parity-check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths. Our approach to designing high-rate quantum error-correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes, making it possible to deal with a situation where promising quantum analogues of low-density parity-check codes are difficult to find.

关键词： quantum error correction low-density parity-check code combinatorial design entanglement-assisted quantum error-correcting code

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Linear codes with arbitrary dimensional hull and their applications to EAQECCs

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quantum INFORMATION PROCESSING 2022年第2期21卷 1-30页

作者： Sok, Lin Qian, Gang Anhui Univ Sch Math Sci Hefei 230601 Peoples R China

The Euclidean hull dimension of a linear code is an important quantity to determine the parameters of an entanglement-assisted quantum error-correcting code (EAQECC) if the Euclidean construction is applied. In this paper, we study the Euclidean hull of a linear code by means of orthogonal matrices. We provide some methods to construct linear codes over F-pm with hull of arbitrary dimensions. With existence of self-dual bases of F-pm over F-p, we determine a Gray map from F-pm to F-p(m), and from a given linear code over F-pm with one-dimensional hull, we construct, using such a Gray map, a linear code over F-p with m-dimensional hull for all m when p is even and for all m odd when p is odd. Comparisons with classical constructions are made, and some good EAQECCs over F-q, q = 2, 3, 4, 5, 9, 13, 17, 49 are presented.

关键词： Hull Optimal code Orthogonal matrix Matrix product code Self-dual basis Gray map entanglement-assisted quantum error-correcting code

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entanglement-assisted quantum codes from arbitrary binary linear codes

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DESIGNS codeS AND CRYPTOGRAPHY 2015年第1期77卷 193-202页

作者： Qian, Jianfa Zhang, Lina Anhui Univ Sci & Technol Coll Sci Huainan 232007 Peoples R China

It is possible to construct an entanglement-assisted quantum error-correcting (EAQEC, for short) code from any classical linear code. However, the parameter of ebits is usually calculated by computer search. In this work, we can construct a family of EAQEC codes from arbitrary binary linear codes, where the parameter of ebits can be easily generated algebraically and not by computational search. Moreover, the constructed EAQEC codes are maximal-entanglement EAQEC codes. We also present a different method of constructing entanglement-assisted accumulator codes. Finally, we prove that asymptotically good EAQEC codes exist.

关键词： entanglement-assisted quantum error-correcting code quantum errorcorrecting code Linear code Parity-check matrix

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On Linear codes With One-Dimensional Euclidean Hull and Their Applications to EAQECCs

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IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第7期68卷 4329-4343页

作者： Sok, Lin Anhui Univ Sch Math Sci Hefei 230601 Anhui Peoples R China

The Euclidean hull of a linear code C is the intersection of C with its Euclidean dual C-perpendicular to. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented.

关键词： Hull MDS code almost MDS code optimal code algebraic curve algebraic geometry code entanglement-assisted quantum error-correcting code

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A new construction of linear codes with one-dimensional hull

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DESIGNS codeS AND CRYPTOGRAPHY 2022年第12期90卷 2823-2839页

作者： Sok, Lin Anhui Univ Sch Math Sci Hefei 230601 Anhui Peoples R China Royal Univ Phnom Penh Dept Math Phnom Penh 12156 Cambodia

The hull of a linear code C is the intersection of C with its dual C-perpendicular to. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The hull of linear codes has recently found its application to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, we provide a new method to construct linear codes with one-dimensional hull. This construction method improves the code lengths and dimensions of the recent results given by the author. As a consequence, we derive several new classes of optimal linear codes with one-dimensional hull. Some new EAQECCs are presented.

关键词： Hull self-orthogonal code MDS code almost MDS code optimal code algebraic curve algebraic geometry code entanglement-assisted quantum error-correcting 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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On the Construction of Hermitian Self-Orthogonal codes Over F9 and Their Application

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INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS 2024年第9期63卷 1-17页

作者： Li, Zhihao Li, Ruihu Guan, Chaofeng Song, Hao Air Force Engn Univ Fundamentals Dept Xian 710051 Shaanxi Peoples R China Henan Key Lab Network Cryptog Technol Zhengzhou 450001 Henan Peoples R China

We construct a lot of optimal or near-optimal [n, k, d](9 )Hermitian self-orthogonal codes for k <= 3 using norm codes and matrix combinatorial construction method. As an application, we construct nine families of entanglement-assisted quantum error-correcting codes. Some of these codes can achieve q-ary linear EA-Griesmer bound with better parameters than those in the literature

关键词： Hermitian self-orthogonal code entanglement-assisted quantum error-correcting code q-ary linear EA-Griesmer bound

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Two Class of entanglement-assisted quantum codes from Shortened Hamming codes

Two Class of Entanglement-Assisted Quantum Codes from Shorte...

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IEEE International Conference on Signal and Image Processing (ICSIP)

作者： Qian, Jianfa Zhang, Lina Huizhou Univ Dept Math Huizhou 516007 Guangdong Peoples R China

ISBN: (纸本)9781509023776

entanglement-assisted quantum error-correcting codes are not only theoretically interesting, but also of great importance in practical physically applications. In this work, two class of entanglement-assisted quantum codes are constructed. The first class of entanglement-assisted quantum codes is minimal ebits entanglement-assisted quantum codes, which require only one copy of maximally entangled state no matter how large the code length is. The second class of entanglement-assisted quantum codes is maximal entanglement entanglement-assisted quantum codes, which can achieve the entanglement-assisted hashing bound asymptotically.

关键词： entanglement-assisted quantum error-correcting code linear code maximal entanglement entanglement-assisted quantum codes

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σ-LCD codes over finite chain rings

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DESIGNS codeS AND CRYPTOGRAPHY 2020年第4期88卷 727-746页

作者： Liu, Xiusheng Liu, Hualu Hubei Polytech Univ Sch Math & Phys Huangshi 435003 Hubei Peoples R China Hubei Univ Technol Sch Sci Wuhan 430068 Hubei Peoples R China

In this work, we first generalize the sigma-LCD codes over finite fields to sigma-LCD codes over finite chain rings. Under suitable conditions, linear codes over finite chain rings that are sigma-LCD codes are characterized. Then we provide a necessary and sufficient condition for free constacyclic codes over finite chain rings to be sigma-LCD. We also get some new binary LCD codes of different lengths which come from Gray images of constacyclic sigma-LCD codes over F-2+ gamma F-2+ gamma F-2(2). Finally, for special finite chain rings F-q + gamma F-q, we define a new Gray map Phi from (F-q + gamma F-q)(n) to F-q(2n), and by using sigma--LCD codes over finite chain rings F-q + gamma F-q, we construct new entanglement-assisted quantum error-correcting (abbreviated to EAQEC) codes with maximal entanglement and parts of them are MDS EAQEC codes.

关键词： Finite chain ring Generator matrix sigma-LCD code entanglement-assisted quantum error-correcting code

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Some new families of entanglement-assisted quantum MDS codes derived from negacyclic codes

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quantum INFORMATION PROCESSING 2022年第9期21卷 318-318页

作者： Wang, Liqi Wang, Pan Zhu, Shixin Hefei Univ Technol Sch Math Hefei 230601 Peoples R China

entanglement-assisted quantum error-correcting codes as a generalization of stabilizer quantum error-correcting (QEC) codes can improve the performance of stabilizer QEC codes and can be constructed from arbitrary classical linear codes by relaxing the dual-containing condition and using pre-shared entanglement states between the sender and the receiver. In this paper, we construct some families of entanglement-assisted quantum maximum distance separable codes with parameters [[q(2)-1/a, q(2)-1/a-2(d - 1)+ c, d;c]](q), where q is an odd prime power with the form q = am +/- l, a = l(2) - 1 or a = l(2)-1/2, l is an odd integer, and m is a positive integer. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature.

关键词： entanglement-assisted quantum error-correcting code Negacyclic code Cyclotomic coset Defining set

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