Article
Computer Science, Information Systems
Xiaoqiang Wang, Chunming Tang, Cunsheng Ding
Summary: The interplay between coding theory and combinatorial t-designs has been a popular topic of research among combinatorialists and coding theorists for many years. While infinite families of cyclic codes supporting 3-designs have been constructed, no infinite family of negacyclic codes supporting 3-designs has been reported. This paper aims to present an infinite family of cyclic codes and two infinite families of negacyclic codes that support 3-designs. The parameters and weight distributions of these codes are determined, and the subfield subcodes of the negacyclic codes over GF(q) are studied. Three infinite families of almost MDS codes and a constacyclic code supporting a 4-design are also presented.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2023)
Article
Telecommunications
Xiaoxiao Li, Minjia Shi
Summary: By utilizing a triple of simplicial complexes, trace codes over a cubic ring extension of the binary field are constructed. A linear Gray map produces multiple infinite families of binary few-weight codes with up to 9 weights, and equating two complexes in the triple leads to minimal and distance optimal binary three-weight codes.
IEEE COMMUNICATIONS LETTERS
(2021)
Article
Computer Science, Information Systems
Stefka Bouyuklieva, Javier de la Cruz
Summary: The aim of this work is to study the structure and properties of binary LCD codes with an automorphism of odd prime order and present a method for their construction.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Telecommunications
Shudi Yang
Summary: By utilizing a proper defining set, a class of linear codes with two or three nonzero weights has been defined and their complete weight enumerators and weight enumerators have been determined using Weil sums. Some of these codes are optimal with respect to the Griesmer bound or Markus Grassl's code tables, making them suitable for applications in strongly regular graphs and secret sharing schemes.
IEEE COMMUNICATIONS LETTERS
(2021)
Article
Mathematics, Applied
Xiaoxiao Li, Minjia Shi
Summary: In this paper, we construct several infinite families of codes over the chain ring R = F-2[u]/< u(k)>, and compute the homogeneous weight distributions of these codes when simplicial complexes are generated by a single maximal element. Through the Gray map, it is determined that some codes are minimal while others are distance optimal. These codes have minimal codewords for inclusion of supports, making them suitable for secret sharing schemes.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2022)
Article
Computer Science, Theory & Methods
Xiaoshan Kai, Jiayuan Zhang, Ping Li, Shixin Zhu
Summary: Two new methods for constructing self-orthogonal codes from known self-orthogonal codes are proposed in this paper. Based on these methods, four infinite classes of binary self-orthogonal codes are constructed. The weight distributions and minimum distances of their dual codes are also determined. Additionally, a class of optimal linear codes and a class of almost optimal linear codes with respect to the Sphere Packing Bound are presented.
CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES
(2023)
Article
Mathematics, Applied
Xiaomeng Zhu, Yangjiang Wei
Summary: In this paper, quaternary linear codes are constructed using simplicial complexes and their weight distributions are determined. Additionally, an infinite family of minimal quaternary linear codes that meet the Griesmer bound is presented.
Article
Mathematics, Applied
Hatoon Shoaib
Summary: This paper studies a special class of quasi-cyclic codes, called double circulant codes, over F4, which have complementary-duals. The main techniques used in this study include Chinese reminder theory decomposition, explicit enumeration, and asymptotics. Particularly, it is shown that the considered class of codes is asymptotically good.
Article
Computer Science, Information Systems
Hai Q. Dinh, Jamal Laaouine, Mohammed E. Charkani, Warattaya Chinnakum
Summary: This paper investigates the Hamming distance of a certain class of cyclic codes over a specific ring and identifies all maximum distance separable codes among them.
Article
Computer Science, Information Systems
Zhao Hu, Nian Li, Xiangyong Zeng, Lisha Wang, Xiaohu Tang
Summary: In this paper, we constructed four families of linear codes that can produce infinite families of optimal linear codes, including (near) Griesmer codes. The optimality of these codes was characterized using the Griesmer bound, and many distance-optimal linear codes were obtained. In-depth discussions on special cases of these families revealed several classes of (distance-) optimal linear codes with few weights.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Computer Science, Theory & Methods
Vidya Sagar, Ritumoni Sarma
Summary: This article studies linear codes over the commutative non-unital ring of size four, including their Lee-weight distributions and binary Gray images. Under certain conditions, these classes of codes are minimal and self-orthogonal, and most of them are few-weight codes.
DESIGNS CODES AND CRYPTOGRAPHY
(2023)
Article
Mathematics, Applied
Vidya Sagar, Ritumoni Sarma
Summary: This paper investigates additive complementary dual (ACD) codes over the ring Z(2)R and explores their properties. Conditions for an additive code to be an ACD code are established, and necessary and sufficient conditions for a separable additive code to be an ACD code are obtained. The paper also studies a Gray map that transforms certain additive codes into binary linear complementary dual (LCD) codes and presents several optimal (or almost optimal) binary LCD codes. Additionally, weight enumerators are computed and the corresponding MacWilliams identities are discussed.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Mathematics
Stefka Bouyuklieva, Iliya Bouyukliev
Summary: A modified version of the Brouwer-Zimmermann algorithm is introduced to calculate the minimum weight of a linear code over a finite field, reducing the number of codewords to consider. It is especially significant when the length of a code is not divisible by its dimensions. This algorithm also has the capability to find all codewords of weight less than a given constant, and has been implemented in the software package QextNewEdition.
Article
Mathematics, Applied
Minjia Shi, Xiaoxiao Li
Summary: In this paper, we construct new families of codes using simplicial complexes and compute their Lee weight distributions. Through the Gray map, we demonstrate that some of these codes are minimal and distance optimal.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Mathematics, Applied
Dev Akre, Nuh Aydin, Matthew J. Harrington, Saurav R. Pandey
Summary: One of the most important and challenging problems in coding theory is to construct codes with optimal parameters and properties. By implementing a fast cyclic partitioning algorithm and the highly effective ASR algorithm, we have discovered 113 new binary quasi-cyclic (QC) codes that have the same parameters as the best known linear codes. These codes also have additional desirable properties such as reversibility, lowest distance codes, self-orthogonality, or dual-containing. Furthermore, we introduce an algorithm for generating new codes from QC codes using ConstructionX and present 33 new record breaking linear codes over GF(2), GF(3), and GF(5) produced by this method.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Computer Science, Theory & Methods
Yansheng Wu, Qin Yue, Xueying Shi, Xiaomeng Zhu
CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES
(2020)
Article
Mathematics
Yongfeng Niu, Qin Yue, Yansheng Wu
DISCRETE MATHEMATICS
(2020)
Article
Computer Science, Information Systems
Fengwei Li, Qin Yue, Yansheng Wu
IEEE TRANSACTIONS ON INFORMATION THEORY
(2020)
Article
Computer Science, Theory & Methods
Yanyan Gao, Qin Yue, Yansheng Wu
DESIGNS CODES AND CRYPTOGRAPHY
(2020)
Article
Mathematics
Yanyan Gao, Qin Yue, Yansheng Wu
Summary: The authors provide precise descriptions and enumerations of linear complementary dual (LCD) codes and self-orthogonal codes in the finite dihedral group algebras F-q [D-2n], with numerical examples to illustrate the main results.
CHINESE ANNALS OF MATHEMATICS SERIES B
(2021)
Article
Computer Science, Information Systems
Yansheng Wu, Jong Yoon Hyun, Yoonjin Lee
Summary: This paper constructs new classes of Euclidean LCD MDS codes and Hermitian LCD MDS codes that are not monomially equivalent to Reed-Solomon codes, known as LCD MDS codes of non-Reed-Solomon type. The method is based on previous constructions by Beelen et al. (2017) and Roth and Lempel (1989). This is the first paper on non-Reed-Solomon type LCD MDS codes, ensuring uniqueness and innovation.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2021)
Article
Computer Science, Theory & Methods
Yansheng Wu, Yoonjin Lee, Qiang Wang
Summary: This paper presents further improvement of index bounds for character sums of polynomials over finite fields, demonstrating examples that show the new bound is superior to existing bounds. An application of this improvement is an estimation of the number of solutions of algebraic curves.
DESIGNS CODES AND CRYPTOGRAPHY
(2022)
Article
Computer Science, Information Systems
Yansheng Wu, Chengju Li, Fu Xiao
Summary: In this paper, we investigate the properties of quaternary linear codes and their binary subfield codes. We establish a relationship between quaternary linear codes and their binary subfield codes, and examine the weight distribution of these codes through the construction of models. We also present results regarding optimal codes and nearly optimal codes.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Mathematics
Yansheng Wu, Jong Yoon Hyun, Yoonjin Lee
Summary: The authors provide an explicit criterion, based on the Walsh spectrum, to determine whether p-ary functions can produce association schemes. Using this criterion, they establish a correlation between p-ary bent functions and association schemes, showing that a p-ary bent function induces a p-class association scheme only if it is weakly regular. They apply their main criterion to construct numerous few-class association schemes from p-ary functions and present four classes of p-ary two-weight linear codes derived from the association schemes.
JOURNAL OF COMBINATORIAL THEORY SERIES A
(2022)
Article
Mathematics, Applied
Yansheng Wu, Chengju Li, Shangdong Yang
Summary: In this paper, the Euclidean hulls and general Galois hulls of generalized Reed-Solomon codes are investigated. It is proven that the Galois hulls of certain GRS codes remain GRS codes. Examples of Galois LCD and self-dual MDS codes are also provided. Compared to known results, the Galois hulls of GRS codes obtained in this work have flexible parameters.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Mathematics, Applied
Yinzhao Lei, Chengju Li, Yansheng Wu, Peng Zeng
Summary: This paper explores the hull of linear codes and its applications in communication and cryptography. By using the defining set of the code, a general characterization of the hull's dimension is presented. The focus is mainly on primitive q-ary BCH codes, and sufficient and necessary conditions for the dimension of the hulls are given. Additionally, several classes of self-orthogonal codes are proposed and their parameters are investigated.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Computer Science, Theory & Methods
Yansheng Wu, Jong Yoon Hyun, Qin Yue
Summary: This paper explores the construction of optimal binary linear codes using simplicial complexes and arbitrary posets. By utilizing the posets of two chains, four classes of optimal binary linear codes are obtained, two of which are Griesmer codes that are not equivalent to those constructed by Belov. These codes are also applied in cryptography for secret sharing schemes.
ADVANCES IN MATHEMATICS OF COMMUNICATIONS
(2022)
Article
Mathematics, Applied
Yansheng Wu, Qin Yue
Summary: This study focuses on the irreducible factorization of x(n) - 1 over the finite field F-q, and calculates the number of irreducible factors when the order of q modulo rad(n) is a product of two primes.
DISCRETE MATHEMATICS ALGORITHMS AND APPLICATIONS
(2021)
Article
Telecommunications
Yansheng Wu
Summary: The hull of a linear code is the intersection of the code and its dual. Algorithms for checking permutation equivalence of linear codes and computing the automorphism group are effective when the hull is small. MDS codes meet the Singleton bound, and twisted Reed-Solomon codes are a generalization that is useful for constructing MDS codes. Twisted Reed-Solomon MDS codes with one-dimensional hulls have been obtained in this letter, which are not monomially equivalent to Reed-Solomon codes.
IEEE COMMUNICATIONS LETTERS
(2021)
