Article
Mathematics, Applied
Alessio Meneghetti, Marco Pellegrini, Massimiliano Sala
Summary: The research proposes a new set of linear relations that must be satisfied by the coefficients of the weight distribution. These relations help derive known identities for interesting cases, such as extremal codes, Hermitian codes, MDS, and NMDS codes, in an easier way. Additionally, the weight distribution of AMDS codes is presented for the first time, along with a discussion on the link between the results and the Pless equations.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Engineering, Electrical & Electronic
Mao-Ching Chiu, Yi-Sheng Su
Summary: Ultra-reliable low-latency communication (URLLC) is an important feature of 5G NR. This paper proposes permutation-coded modulations for URLLC, which achieve remarkable performance levels with low-rate and short-block-length applications. To reduce complexities, a trapezoidal permutation-coded modulation scheme is proposed, which can be efficiently decoded by SCL decoders and achieves dispersion bounds for short block lengths.
IEEE TRANSACTIONS ON COMMUNICATIONS
(2023)
Article
Mathematics, Applied
Mevlut Tekkoyun, Ergun Yaraneri
Summary: This article studies linear codes over the F-q-algebra F-q x (F-q + vF(q)) of order q(3), where v(2) = v and F-q is a finite field of q elements. The research not only generalizes most of the existing results for q = 2, but also introduces some new findings. The work is comprehensive, covering standard forms of generator matrices, free codes, dual codes, and cyclic codes.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Mathematics
Lingyong Ma, Guanjun Li, Fengyan Liu
Summary: This paper focuses on linear codes with few weights and constructs more 3-weight linear codes using ternary near-bent and 2-plateaued functions or r-ary functions, where r is a prime. The weight distributions of the resulting linear codes are determined using some exponential sums.
JOURNAL OF MATHEMATICS
(2021)
Article
Telecommunications
Hanqi Tang, Zhe Zhai, Qifu Tyler Sun, Xiaolong Yang
Summary: Permutation linear network coding (LNC) is a generalized version of circular-shift LNC, which offers more linear coding operations for efficient implementation. We prove that a multicast network has an L-dimensional permutation linear solution over a ring R if and only if it has a scalar linear solution over R. This implies that the capacity of a multicast network not solvable by scalar linear solution over R cannot be achieved by permutation LNC either. Moreover, we demonstrate the advantage of permutation LNC over circular-shift LNC in terms of shorter block length for generating a linear solution at a rate smaller than 1.
IEEE COMMUNICATIONS LETTERS
(2023)
Article
Computer Science, Theory & Methods
Basri Caliskan, Nuh Aydin, Peihan Liu
Summary: In this paper, the skew-cyclic codes over the ring S = Z(4) + uZ(4) + vZ(4) with certain properties are studied. The structural properties of the skew polynomial ring S[x, ?], where ? is an automorphism of S, are discussed. The generator and parity-check matrices of ?-cyclic codes over S are determined and their Gray images are shown to be free linear codes over Z(4). Furthermore, double skew-cyclic codes are introduced and their Gray images result in new linear codes over Z(4).
CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES
(2023)
Article
Computer Science, Theory & Methods
Jian Gao, Xiangrui Meng, Fang-Wei Fu
Summary: This paper determines the weight distribution of several classes of double cyclic codes over Galois rings using Gauss sums.
DESIGNS CODES AND CRYPTOGRAPHY
(2022)
Article
Computer Science, Theory & Methods
Zihan Zhang
Summary: Researchers introduce a new metric closely related to the block permutation metric and use it to construct codes. They propose an algebraic-geometric construction based on this metric and improve previous results by constructing non-systematic codes and providing an explicit and systematic construction in the block permutation metric.
DESIGNS CODES AND CRYPTOGRAPHY
(2023)
Article
Computer Science, Information Systems
Kangquan Li, Chunlei Li, Tor Helleseth, Longjiang Qu
Summary: In this paper, the application of two-to-one functions in two constructions of binary linear codes is discussed, resulting in various classes of linear codes with different nonzero weights. The weight distributions of the proposed codes with one weight and with three weights are determined, and the minimum distance of the constructed codes is analyzed, showing that some of them achieve the sphere packing bound. Additionally, examples demonstrate that some of the codes in this paper have best-known parameters.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2021)
Article
Computer Science, Theory & Methods
Youcef Maouche
Summary: In this paper, the symbol-pair weight distribution of irreducible cyclic codes is studied. Generalized cyclotomic numbers are defined and their properties are provided. By using a character sum, a relationship between the symbol-pair weight of irreducible cyclic codes, generalized cyclotomic numbers, and Gaussian periods is established. Moreover, the symbol-pair weight distribution of certain classes of irreducible cyclic codes is determined.
CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES
(2023)
Article
Mathematics, Applied
Shudi Yang, Zheng-An Yao
Summary: A class of projective binary linear codes with at most three nonzero weights are constructed and their weight distributions are investigated using Weil sums. Some of these codes contain optimal codes, and their dual codes are also studied, with some being optimal or almost optimal.
Article
Mathematics, Applied
Yunfei Su, Xiaoshan Kai
Summary: Cyclic codes with two zeros and length n=q(m-1)/r are explored in this study. A necessary and sufficient condition with minimum distance two is provided for such cyclic codes. It is proven that these cyclic codes have a minimum distance of at most three when q>2r+1. An optimal family of cyclic codes with parameters [2(q(m)-1)/q-1, 2(q(m)- 1)/ q-1 - 2m, 4] is obtained for the case of r=q-1/2.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Computer Science, Hardware & Architecture
Xina Zhang, Xiaoni Du, Rong Wang, Fujun Zhang
Summary: This paper presents a class of five-weight linear codes over odd prime field, which have various applications in different fields and are close to being optimal in some aspects.
IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES
(2021)
Article
Computer Science, Theory & Methods
Can Xiang, Chunming Tang, Qi Liu
Summary: This paper investigates an infinite family of antiprimitive cyclic codes and their connections to combinatorial designs, demonstrating that these codes and their dual can construct some infinite families of 3-designs with properties of Steiner systems.
DESIGNS CODES AND CRYPTOGRAPHY
(2022)
Article
Mathematics, Applied
Emily Bergman, Robert S. Coulter, Irene Villa
Summary: By utilizing Hermite's criteria, we were able to classify planar monomials over fields of order a prime cubed, thus confirming the DembowskiOstrom conjecture for monomials over fields of such orders.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Computer Science, Software Engineering
Guang Yang, Chunlei Li, Kjell E. Marstein
Summary: This paper introduces a blockchain-based architecture for current electronic health record systems, aiming to ensure data integrity, improve system interoperability, and introduce a new incentive mechanism. The architecture is independent of specific blockchain platforms and can potentially fit with other electronic record systems requiring protection against data misuse.
CONCURRENCY AND COMPUTATION-PRACTICE & EXPERIENCE
(2021)
Article
Computer Science, Information Systems
Chunming Tang, Cunsheng Ding
Summary: This paper resolves the longstanding question by introducing an infinite family of BCH codes and a linear code family holding spherical geometry designs, paving the way for new research directions in searching for t-designs using elementary symmetric polynomials.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2021)
Article
Computer Science, Theory & Methods
Cunsheng Ding, Chunming Tang
Summary: This paper focuses on studying the linear codes of t-designs held in the Reed-Muller and Simplex codes, presenting some general theory for linear codes containing t-designs and introducing several open problems for further research.
CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES
(2021)
Article
Computer Science, Information Systems
Yang Liu, Cunsheng Ding, Chunming Tang
Summary: The puncturing technique has achieved significant progress over the past 70 years, resulting in many families of linear codes with interesting parameters, while research on the shortening technique remains limited. This paper presents eleven families of optimal shortened codes over finite fields and constructs five infinite families of 2-designs as a byproduct of the study.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2021)
Article
Computer Science, Information Systems
Binkai Gong, Cunsheng Ding, Chengju Li
Summary: This paper investigates the symmetry problem of BCH codes, provides the symmetry conditions for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes, and studies their dual codes. For binary primitive narrow-sense BCH codes, improved bounds on the minimum distances of the dual codes are obtained, and the question of which subclasses of cyclic codes are BCH codes is answered to some extent.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Computer Science, Information Systems
Can Xiang, Chunming Tang, Cunsheng Ding
Summary: This paper investigates two families of linear codes from APN functions and some p-ary shortened codes associated with PN functions, and determines the weight distributions of these shortened codes and the parameters of their duals. The results indicate that the shortening technique has great potential for constructing good codes.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
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, Information Systems
Qi Liu, Cunsheng Ding, Sihem Mesnager, Chunming Tang, Vladimir D. Tonchev
Summary: This paper studies some q-ary BCH codes with length q + 1, focusing on narrow-sense antiprimitive BCH codes. By using tools from algebraic coding theory, combinatorial designs, and group theory, the dimension, minimum distance, and dual codes of these BCH codes are determined.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Computer Science, Information Systems
Kangquan Li, Yue Zhou, Chunlei Li, Longjiang Qu
Summary: This paper presents two new families of APN functions, one in bivariate form and one in L(z) form. By calculating the G-rank, it is demonstrated that these two families are not equivalent to known families and cover known sporadic APN instances.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Computer Science, Information Systems
Haode Yan, Yongbo Xia, Chunlei Li, Tor Helleseth, Maosheng Xiong, Jinquan Luo
Summary: This paper investigates the differential spectrum of the mapping x(pn-3) over F-pn by studying quadratic character sums and equations, providing a unified approach for any odd prime p. The results show that the differential spectrum can be explicitly expressed in terms of n for any given odd prime p, with a special elliptic curve over F-p playing a significant role in the computation for the general case p >= 5.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Computer Science, Information Systems
Chunming Tang, Qi Wang, Cunsheng Ding
Summary: This paper studies the quaternary subfield subcodes and quaternary subfield codes of a subfamily of MDS codes for even m. A family of quaternary cyclic codes is obtained, which are distance-optimal in some cases and generally very good. Furthermore, two infinite families of 3-designs from these quaternary codes and their duals are presented.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Computer Science, Information Systems
Chunming Tang, Cunsheng Ding
Summary: This paper introduces the binary quadratic-residue codes and the punctured Reed-Muller codes R.2((ru-1)/2, m)), two families of binary cyclic codes with special parameters and minimum distance bounds. The objective of the paper is to construct two families of binary cyclic codes with length 2^m-1 and dimension near 2m-1.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
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
Computer Science, Information Systems
Mengyuan Fan, Chengju Li, Cunsheng Ding
Summary: As a special subclass of cyclic codes, BCH codes are among the best cyclic codes and are widely used in communication, storage systems, and consumer electronics. This paper aims to derive a necessary and sufficient condition for two classes of narrow-sense BCH codes to be Hermitian dually-BCH codes and improve the lower bounds on the minimum distances of their Hermitian dual codes.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2023)
Proceedings Paper
Computer Science, Interdisciplinary Applications
Cunsheng Ding, Zhonghua Sun, Xiaoqiang Wang
Summary: Constacyclic codes are a family of linear codes that include cyclic codes as a subclass. They are of theoretical importance and outperform cyclic codes in several aspects. In practice, constacyclic codes are important due to their rich algebraic structures and potential for efficient decoding algorithms. This extended abstract presents the construction of two classes of constacyclic codes using a general construction with cyclic codes, analyzes their parameters, and discusses some open problems.
ARITHMETIC OF FINITE FIELDS, WAIFI 2022
(2023)
Article
Mathematics
Weijun Fang, Jiejing Wen, Fang-Wei Fu
Summary: This paper proposes a sufficient condition to ensure that a Hermitian self-orthogonal GRS code is still a Hermitian self-orthogonal code. Based on this, a new general construction of infinitely families of quantum MDS codes is presented, and two new constructions of quantum MDS codes with flexible parameters are given using the trace function and norm function over finite fields. The constructed quantum MDS codes have different lengths from previous known results, and the minimum distances of all the q-ary quantum MDS codes are larger than q/2 + 1.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Hoa T. Bui, Guillermo Pineda-Villavicencio, Julien Ugon
Summary: The paper examines the linkedness of the graphs of cubical polytopes and proves that every cubical d-polytope has [d/2] and strong [d/2] linkedness, for every d = 3. These results are optimal for this class of polytopes.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Julia Carrigan, Isaiah Hollars, Eric Rowland
Summary: Two words p and q are avoided by the same number of length-n words, for all n, precisely when p and q have the same set of border lengths. Previous proofs of this theorem use generating functions but do not provide an explicit bijection. We give a bijective proof for all pairs p, q that have the same set of proper borders, establishing a natural bijection from the set of words avoiding p to the set of words avoiding q.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Luke Nelson, Kevin Treat
Summary: We define a poset called the Outer Tamari poset, which is shown to be isomorphic to a subposet of the Tamari lattice introduced by Pallo (1986) and further studied as the Comb poset by Csar, Sengupta, and Suksompong (2014). By using the Outer Tamari poset, we develop recursive formulas for the number of triangulations of the 3-dimensional cyclic polytopes. These triangulations can be considered as elements of both the higher Stasheff-Tamari orders in dimension three and the Tamari Block lattices defined in a previous article. Therefore, our work here can be seen as constructing recursive enumerations of these posets.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Nicholas A. Loehr, Gregory S. Warrington
Summary: This study explores variants of chromatic symmetric functions for rooted graphs and investigates the combinatorial identities and recursions satisfied by these rooted chromatic polynomials. It proves the irreducibility of Stanley's polynomial under certain conditions, establishes conditions for isomorphism of rooted trees as rooted graphs, and provides a combinatorial interpretation of the monomial expansion of pointed chromatic functions.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Shira Zerbib
Summary: This article studies the property of a family of sets and proves that when a family of compact convex sets has a specific intersection property, it can be pierced by a certain number of lines. The proofs are based on the topological KKM theorem.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
David Sossa, Vilmar Trevisan
Summary: The study focuses on the complementarity spectrum and separability index of graphs, demonstrating the relationship between the largest complementarity eigenvalues of graphs of a specific order and deducing the growth trend of the separability index of the set of connected graphs.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Anita Keszler, Zsolt Tuza
Summary: This study considers edge decompositions of K-v((3)) - I and provides decomposition results satisfying certain conditions, complementing previous research findings.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Ping Li
Summary: This paper explores the relationship between monochromatic connection coloring and the connectivity of a graph, providing a method and upper bounds for computing the monochromatic connection number. Additionally, the paper discusses the characteristics of MC-colorings for graphs with specific connectivity requirements.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Santiago Guzman-Pro
Summary: This article discusses the concepts of full-homomorphism, full H-colouring, and minimal H-obstruction. It proves the existence of a finite number of minimal H obstructions for every graph H. Furthermore, it describes the properties of minimal obstructions and poses some related questions.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Rongzhong Xiao
Summary: We prove that for any finite coloring of Q, there exist non-zero elements that satisfy certain conditions.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Robert Lukot'ka
Summary: The article studies the circular flow problem, gives the circular flow number of Goldberg snark G2k+1, and proves a conjecture.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Benjamin Egan, Yuri Nikolayevsky
Summary: A simple graph is called triangular if every edge of it belongs to a triangle. We conjecture that any graphical degree sequence all terms of which are greater than or equal to 4 has a triangular realisation, and establish this conjecture for a class of biregular graphical degree sequences.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Jakub Przybylo
Summary: We improve upon Molloy's breakthrough result by adapting Bernshteyn's proof, achieving a stronger result that states triangle-free graphs can be colored from lists of size (1 +o(1))A/ log A, where vertices sharing a common color do not induce a triangle in G. We also extend this result to graphs of maximum degree A by proving the sufficiency of list sizes (1000 + o(1))A/ log A, as implied by a more general result due to Amini and Reed. Furthermore, we demonstrate that lists of length 2(r - 2)A log2 log2 A/ log2 A are sufficient if one replaces the triangle with any Kr with r >= 4. All of these bounds hold in the context of correspondence colorings.
DISCRETE MATHEMATICS
(2024)
Article
Mathematics
Hongwei Zhu, Minjia Shi
Summary: This paper studies the b-symbol weight hierarchy of Kasami codes and discusses their applications in high density data storage systems.
DISCRETE MATHEMATICS
(2024)
