Article
Mathematics, Applied
Xiaoru Li, Ziling Heng
Summary: This paper presents several constructions of near maximum distance separable (NMDS) codes and determines their weight enumerators. It also determines the locality of these codes and obtains many families of distance-optimal and dimension-optimal locally recoverable codes.
FINITE FIELDS AND THEIR APPLICATIONS
(2023)
Article
Engineering, Electrical & Electronic
Gaojun Luo, Xiwang Cao
Summary: This paper focuses on constructing optimal binary locally recoverable codes meeting the alphabet-dependent bound, using a general framework for linear codes associated to a set. The problem of designing optimal binary locally recoverable codes is turned into constructing a suitable set, resulting in several constructions of these codes. Finally, the paper proposes constructions of optimal binary locally recoverable codes with locality 2 and locality parameters (r, delta) by Griesmer codes.
IEEE TRANSACTIONS ON COMMUNICATIONS
(2021)
Article
Computer Science, Theory & Methods
Angela Aguglia, Luca Giuzzi, Angelo Sonnino
Summary: The paper presents a geometric construction of [n, 9, n - 9](q) near-MDS codes derived from elliptic curves with n F-q-rational points. It also demonstrates that in certain cases, these codes cannot be extended to longer near-MDS codes.
DESIGNS CODES AND CRYPTOGRAPHY
(2021)
Article
Mathematics
Qiuyan Wang, Ziling Heng
Summary: MDS codes have parameters [n, k, n - k + 1], while almost MDS codes have parameters [n, k, n-k] and both the code and its dual are almost maximum distance separable. Near MDS codes are interesting objects in finite geometry and constructing an infinite family with settled weight distributions is challenging.
DISCRETE MATHEMATICS
(2021)
Article
Computer Science, Information Systems
Ziling Heng, Chengju Li, Xinran Wang
Summary: This paper aims to construct several infinite families of MDS, near MDS and almost MDS codes from some special cyclic subgroups of F-q2* and study the minimum linear locality of these codes. Some of the codes are also suitable for error detection.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Computer Science, Theory & Methods
Li Xu, Zhengchun Zhou, Jun Zhang, Sihem Mesnager
Summary: This paper studies the optimal (r, delta)-LRCs over quaternary field and proposes their classification and explicit code constructions by examining all possible cases.
DESIGNS CODES AND CRYPTOGRAPHY
(2022)
Proceedings Paper
Computer Science, Interdisciplinary Applications
Ziling Heng, Xiaoru Li
Summary: In this paper, several infinite families of near MDS codes with dimension four are constructed using special matrixes and oval polynomials. The weight enumerators of these codes are explicitly determined. As an application, the duals of these near NMDS codes are proven to be both distance-optimal and dimension-optimal locally recoverable codes.
ARITHMETIC OF FINITE FIELDS, WAIFI 2022
(2023)
Article
Mathematics
Ziling Heng, Xinran Wang
Summary: Infinite families of NMDS codes holding 2-designs or 3-designs have been constructed by Ding and Tang, marking a breakthrough in this field where only a few known families exist. This paper aims to construct new families of NMDS codes holding t-designs, with different parameters compared to existing codes. The weight enumerators of the NMDS codes are determined, and their 2-designs or 3-designs are proven. Additionally, several infinite families of optimal locally recoverable codes are derived via the NMDS codes.
DISCRETE MATHEMATICS
(2023)
Article
Mathematics, Applied
Boran Kim
Summary: In this study, we focus on locally recoverable codes in Hermitian function fields and provide an explicit formula for the improved lower bound of minimum distance. We also present a criterion for suggesting an improved lower bound based on divisors with certain places.
Article
Mathematics, Applied
Xiaojun Geng, Meng Yang, Jun Zhang, Zhengchun Zhou
Summary: MDS codes and AMDS codes, special classes of linear codes, are widely used in communication, data storage, combinatorial theory, and secret sharing. This paper presents a class of AMDS codes derived from BCH codes and determines their parameters. The proposed AMDS codes are found to be distance-optimal and dimension-optimal locally repairable codes.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Computer Science, Theory & Methods
Michela Ceria, Antonio Cossidente, Giuseppe Marino, Francesco Pavese
Summary: This article describes several classes of near-MDS sets in PG(3,q). They can be obtained either by considering the intersection of an elliptic quadric ovoid and a Suzuki-Tits ovoid in a symplectic polar space W(3,q), or starting from the points of a twisted cubic in PG(3,q). Additionally, two classes of complete caps of size 2q(2) - q +/- root 2q + 2 in PG(4,q) are exhibited.
DESIGNS CODES AND CRYPTOGRAPHY
(2023)
Article
Computer Science, Information Systems
Ziling Heng, Cunsheng Ding
Summary: This paper studies the subfield codes of geometric codes with dimension 3 and 4 over large finite fields and obtains distance-optimal subfield codes. The key idea is to choose good linear codes over extension fields with small dimensions. The results include two families of dimension-optimal codes and several families of nearly optimal codes. Additionally, several open problems are proposed in this paper.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Computer Science, Theory & Methods
Xiaoqiang Wang, Cunsheng Ding, Hongwei Liu, Dabin Zheng
Summary: In this paper, MDS constacyclic codes over finite fields are characterized, constructed, and presented. The paper also provides rationales for distinguishing different classes of codes and summarises some applications of MDS codes in cryptography.
CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES
(2023)
Article
Computer Science, Theory & Methods
Can Xiang, Jinquan Luo
Summary: Subfield codes of linear codes over finite fields, particularly binary subfield codes constructed from a special family of MDS codes, are studied in this paper. The parameters of these codes, as well as their dual codes, are explicitly determined. Some of the codes presented in the paper are proven to be optimal or almost optimal.
ADVANCES IN MATHEMATICS OF COMMUNICATIONS
(2021)
Article
Computer Science, Information Systems
Guangkui Xu, Xiwang Cao, Longjiang Qu
Summary: This paper focuses on the construction of infinite families of 3-designs and 2-designs using special equations and linearized polynomials. These designs possess new and flexible parameters.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
