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Constructions of mds, Near mds and almost mds codes From Cyclic Subgroups of Fq2*

IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第12期68卷 7817-7831页

作者： Heng, Ziling Li, Chengju Wang, Xinran Changan Univ Sch Sci Xian 710064 Peoples R China Southeast Univ Natl Mobile Commun Res Lab Nanjing 210096 Peoples R China East China Normal Univ Shanghai Key Lab Trustworthy Comp Shanghai 200062 Peoples R China

Linear codes achieving or nearly achieving the Singleton bound are interesting in both theory and practice. The objective of this paper is to construct several infinite families of mds, near mds and almost mds codes from some special cyclic subgroups of F-q2*. To this end, the augmentation and extension techniques are used. The codes in this paper have flexible parameters and their lengths could be large. The minimum linear locality of the codes constructed in this paper is also studied. Some infinite families of optimal linearly locally recoverable codes are obtained. Besides, some codes in this paper are proved to be proper for error detection.

关键词： mds code near mds code almost mds code locally recoverable code

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Infinite Families of 3-Designs and 2-Designs From almost mds codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第7期68卷 4344-4353页

作者： Xu, Guangkui Cao, Xiwang Qu, Longjiang Huainan Normal Univ Sch Finance & Math Huainan 232038 Peoples R China Nanjing Univ Aeronaut & Astronaut Dept Math Nanjing 210016 Peoples R China MIIT Key Lab Math Modeling & High Performance Comp Air Nanjing 210016 Peoples R China Natl Univ Def Technol Coll Liberal Arts & Sci Changsha 410073 Peoples R China Hunan Engn Res Ctr Commercial Cryptog Theory & Te Changsha 410073 Peoples R China

Combinatorial designs are closely related to linear codes. Recently, some near mds codes were employed to construct t-designs by Ding and Tang, which settles the question as to whether there exists an infinite family of near mds codes holding an infinite family of t-designs for t >= 2. This paper is devoted to the construction of infinite families of 3-designs and 2-designs from special equations over finite fields. First, we present an infinite family of almost mds codes over GF(p(m)) holding an infinite family of 3-designs. We then provide an infinite family of almost mds codes over GF(p(m)) holding an infinite family of 2-designs for any field GF(q). In particular, some of these almost mds codes are near mds. Second, we present an infinite family of near mds codes over GF(2(m)) holding an infinite family of 3-designs by considering the number of roots of a special linearized polynomial. Compared to previous constructions of 3-designs or 2-designs from linear codes, the parameters of some of our designs are new and flexible.

关键词： Linear code almost mds code near mds code polynomial 3-design 2-design

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Two classes of quantum codes from almost mds codes

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INTERNATIONAL JOURNAL OF QUANTUM INFORMATION 2023年第8期21卷

作者： Sun, Zhonghua Liu, Xinyue Zhu, Shixin Hefei Univ Technol Sch Math Hefei 230601 Anhui Peoples R China

Hermitian dual-containing codes are an important class of linear codes which have important applications in the construction of quantum codes. In this paper, two classes of Hermitian dual-containing almost mds codes over finite fields are studied. By employing the Hermitian construction, a class of quantum codes with minimum distance 3 and a class of quantum codes with minimum distance 4 are constructed.

关键词： Quantum code Hermitian dual-containing code almost mds code cyclic code

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New families of self-dual codes

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DESIGNS codeS AND CRYPTOGRAPHY 2021年第5期89卷 823-841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

Recently, the author has constructed families of mds Euclidean self-dual codes from genus zero algebraic geometry (AG) codes. In the present correspondence, more families of optimal Euclidean self-dual codes from AG codes are explored. New families of mds Euclidean self-dual codes of odd characteristic and those of almost mds Euclidean self-dual codes are constructed explicitly from genus zero and genus one curves, respectively. More families of Euclidean self-dual codes are constructed from algebraic curves of higher genus.

关键词： Self-orthogonal code Self-dual code mds code almost mds code Optimal code Algebraic curve Algebraic geometry code Differential algebraic geometry code

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Projective Linear codes From Some almost Difference Sets

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IEEE TRANSACTIONS ON INFORMATION THEORY 2023年第2期69卷 978-994页

作者： Heng, Ziling Changan Univ Sch Sci Xian 710064 Peoples R China

Projective linear codes are a special class of linear codes whose duals have minimum distance at least 3. The columns of the generator matrix of an [n, k] projective code over finite field F-q can be viewed as points in the projective space PG( k - 1, F-q). Projective codes are of interest not only because their duals have good error correcting capability but also because they may be related to interesting combinatorial structures. The objective of this paper is to construct projective linear codes with five families of almost difference sets. To this end, the augmentation and extension techniques for linear codes are used. The parameters and weight distributions of the projective codes are explicitly determined. Several infinite families of optimal or almost optimal codes including mds codes, near mds codes, almost mds odes and Griesmer codes are obtained. Besides, we also give some applications of these codes.

关键词： codes Linear codes Additives codecs Generators Standards Indexes Linear code almost difference set almost mds code locally recoverable code

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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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New Lower Bounds for Permutation codes Using Linear Block codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第7期66卷 4019-4025页

作者： Micheli, Giacomo Neri, Alessandro Univ S Florida Dept Math Tampa FL 33620 USA INRIA Saclay Ile de France F-91120 Palaiseau France

In this paper we prove new lower bounds for the maximal size of permutation codes by connecting the theory of permutation codes with the theory of linear block codes. More specifically, using the columns of a parity check matrix of an [n,k,d](q) linear block code, we are able to prove the existence of a permutation code in the symmetric group of degree n, having minimum distance at least d and large cardinality. With our technique, we obtain new lower bounds for permutation codes that enhance the ones in the literature and provide asymptotic improvements in certain regimes of length and distance of the permutation code.

关键词： Parity check codes Hamming distance Hamming weight Generators Linear codes Symmetric matrices Permutation code powerline communications linear block code maximum distance separable (mds) code almost mds code permutation array

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