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
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
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
Mathematics, Applied
Yang Pan, Yan Liu
Summary: This article describes two classes of few-weight ternary codes and computes their minimum weight and weight distribution using mathematical objects called simplicial complexes. One class of codes mentioned here has the same parameters as binary first-order Reed-Muller codes. A class of (optimal) minimal linear codes is also obtained in this correspondence.
Article
Mathematics, Applied
Shukai Wang, Minjia Shi
Summary: By selecting the down-set appropriately, the weight distribution of the additive code C-L was determined, leading to several infinite families of minimal and optimal few-weight p-codes obtainable through the Gray map, which are applicable in secret sharing schemes.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2022)
Article
Mathematics, Applied
Minjia Shi, Xiaoxiao Li
Summary: This study focuses on constructing linear codes using down-sets, computing Lee weight distributions of two classes of codes, and obtaining minimal and distance optimal codes through the Gray map.
DISCRETE APPLIED MATHEMATICS
(2021)
Article
Telecommunications
Rongsheng Wu, Minjia Shi
Summary: Mixed alphabet codes are generalizations of classical linear codes over finite fields and rings. The letter introduces a method to construct FpR-additive codes, which are a combination of irreducible cyclic codes over F-p and trace codes over R. By using the Gray map, part of the obtained image codes meets the Greismer bound and can be used to construct new secret sharing schemes.
IEEE COMMUNICATIONS LETTERS
(2021)
Article
Physics, Fluids & Plasmas
Leonie Neuhaeuser, Renaud Lambiotte, Michael T. Schaub
Summary: The study indicates that in network systems with time-dependent, multiway interactions, the convergence speed of consensus dynamics is slower than systems with only pairwise interactions, and slower than consensus dynamics on corresponding static networks. Additionally, the final consensus value in a temporal system may differ significantly from the consensus value on an aggregated, static hypergraph, with early movers having a greater influence.
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
Multidisciplinary Sciences
L. V. Gambuzza, F. Di Patti, L. Gallo, S. Lepri, M. Romance, R. Criado, M. Frasca, V. Latora, S. Boccaletti
Summary: Various systems have been successfully modeled as networks of coupled dynamical systems, with recent studies showing the presence of higher-order many-body interactions in social groups, ecosystems, and the human brain. The proposed analytical approach by Gambuzza et al. provides conditions for stable synchronization in many-body interaction networks.
NATURE COMMUNICATIONS
(2021)
Article
Multidisciplinary Sciences
Huan Wang, Chuang Ma, Han-Shuang Chen, Ying-Cheng Lai, Hai-Feng Zhang
Summary: Researchers have developed a general framework that combines statistical inference and expectation maximization to fully reconstruct the topology of 2-simplicial complexes with two- and three-body interactions based on binary time-series data. The framework's effectiveness has been validated, demonstrating its robustness against noisy data or stochastic disturbance.
NATURE COMMUNICATIONS
(2022)
Article
Mathematics
Mohammad Reza-Rahmati, Gerardo Flores
Summary: This paper introduces the concepts of graded linear resolution and graded linear quotients, and compares them with componentwise linearity. The paper also provides specific characterizations for certain modules and rings, and offers analogous results for the Orlik-Terao ideal of hyperplane arrangements.
JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics, Applied
Y. Lee, J. Lee, S. M. Oh, D. Lee, B. Kahng
Summary: The simplicial complex representation is a mathematical framework for describing the high-order interaction effects of complex groups in various systems. The homological percolation transitions can be determined by the first and second Betti numbers. A minimal SC model with growth and preferential attachment factors successfully reproduces the HPTs in social coauthorship relationships.
Article
Biology
Blake Bordelon, Cengiz Pehlevan
Summary: This study investigates how to learn from a limited number of experiences and explores the implementation and formation of inductive biases in neural codes. The research finds that the structure of population codes and the match between the code and the task play a crucial role in sample-efficient learning, which is validated through recordings from the mouse primary visual cortex.
Article
Mathematics, Applied
Florian Frick, Mirabel Hu, Verity Scheel, Steven Simon
Summary: We provide a simple characterization of simplicial complexes on few vertices that embed into the d-sphere. Namely, a simplicial complex on d + 3 vertices embeds into the d-sphere if and only if its non-faces do not form an intersecting family. As immediate consequences, we recover the classical van Kampen-Flores theorem and provide a topological extension of the Erdos-Ko-Rado theorem. By analogy with Fary's theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.
ANNALS OF COMBINATORICS
(2023)
Article
Mathematics, Applied
Li Xu, Cuiling Fan, Dongchun Han
Summary: This paper investigates near maximum distance separable (NMDS) codes with dimension 3. By adding projective points in specific projective geometries, a new class of NMDS codes is obtained, and their properties are studied.
FINITE FIELDS AND THEIR APPLICATIONS
(2024)
Article
Mathematics, Applied
Shiang Tang
Summary: In this paper, we provide new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is achieved by constructing compatible systems of tadic Galois representations valued in a semisimple group G using Galois theoretic and automorphic methods, and then proving that the Galois images are maximal for a set of primes of positive density based on Larsen's classical result on Galois images for compatible systems.
FINITE FIELDS AND THEIR APPLICATIONS
(2024)
Article
Mathematics, Applied
Huan Sun, Qin Yue, Xue Jia
Summary: In this article, the authors study a family of APN hexanomials F3 that satisfy a certain technical condition. They determine the number of APN hexanomials F3 and provide a theorem for their determination when i = 1. Additionally, they construct a family of APN functions in bivariate form and prove its CCZ-equivalence to F3.
FINITE FIELDS AND THEIR APPLICATIONS
(2024)
Article
Mathematics, Applied
Chandan Kumar Vishwakarma, Rajesh P. Singh
Summary: In this paper, we investigate certain classes of complete permutation polynomials with specific forms and propose methods for constructing PPs and CPPs over finite fields using the AGW criterion. Additionally, we obtain constructions of sets of Mutually orthogonal Latin squares using permutation polynomials over finite fields.
FINITE FIELDS AND THEIR APPLICATIONS
(2024)
Article
Mathematics, Applied
Monika Bishnoi, Pankaj Kumar
Summary: In this paper, we investigate cubic primitive irreducible cyclic codes and provide bounds on their minimum distances. We also demonstrate a connection between solutions of Diophantine equations and weight enumerators of these codes.
FINITE FIELDS AND THEIR APPLICATIONS
(2024)
