Article
Mathematics, Applied
Gyanendra K. Verma, Astha Agrawal, R. K. Sharma
Summary: In this article, a method for constructing multiple Hermitian LCD codes from a given Hermitian LCD code is presented, followed by several methods to construct new Galois LCD codes with different parameters using either a given linear code or a given Galois LCD code. By applying these methods, several new ternary LCD codes with better parameters are constructed for 26 = n = 40 and 21 = k = 30. Additionally, optimal 2-Galois LCD codes over F(2)3 for code length 1 = n = 15 have been obtained. Finally, previous results are extended to the s-inner product from the Euclidean inner product.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Mathematics, Applied
Taher Abualrub, Arezoo Soufi Karbaski, Nuh Aydin, Peihan Liu
Summary: In this paper, we study the structure and properties of additive right and left polycyclic codes induced by a nonbinary vector. We also analyze the relationship between additive polycyclic codes and linear polycyclic codes. Moreover, we construct examples of codes with good parameters.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(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, 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
Marcell Gaal, Robert M. Guralnick
Summary: This paper proves that when a compact simple Lie group G acts absolutely irreducibly on a vector space V, the connected stabilizer of a G-invariant norm almost always coincides with either G or the whole SO(V), with a short list of exceptional cases determined.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Physics, Multidisciplinary
Shikha Yadav, Ashutosh Singh, Om Prakash
Summary: This work investigates skew polycyclic codes over the ring A(e), providing the generator polynomial and generator matrix for these codes using the idempotent decomposition technique. It determines the necessary and sufficient conditions for self-dual codes and derives conditions for skew polycyclic codes to satisfy the complementary duality property. The Gray map of an LCD (or self-dual) code over A(e) is defined and studied, and a construction of LCD codes from self-dual codes is presented. Finally, several examples of entanglement-assisted quantum error-correcting codes obtained from LCD codes are provided.
EUROPEAN PHYSICAL JOURNAL PLUS
(2023)
Article
Environmental Sciences
Hamidreza Shahradnia, Atefeh Chamani, Mehrdad Zamanpoore
Summary: The study utilized an integrated modeling approach to investigate the linkages between landscape-related features and high arsenic levels in river sediments in central Iran. Results showed that sediment arsenic levels were influenced by the density of agricultural activities near the river outlet and the proportion of silty loam or loamy soils, providing important insights for policy decisions regarding river contamination processes in central Iran.
ENVIRONMENTAL SCIENCE AND POLLUTION RESEARCH
(2022)
Article
Mathematics, Applied
Rongsheng Wu, Minjia Shi, Patrick Sole
Summary: Quasi-polycyclic (QP) codes over a finite chain ring R are a generalization of quasi-cyclic codes and can be characterized by their algebraic structure for generator polynomials and minimal generating sets. Using these algebraic properties, new quaternary codes with unique parameters can be derived from 1-generator QP codes using the strong Grobner bases and the Magma system.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2022)
Article
Computer Science, Artificial Intelligence
M. P. Cuellar, J. Gomez-Torrecillas, F. J. Lobillo, G. Navarro
Summary: The minimum distance of linear codes is an NP-hard problem, traditionally solved through algorithm design. This study proves the existence of a column permutation that leads to a simplified row echelon form containing the code distance. Permutations can now be used as a representation scheme, making the search for the optimum in polynomial time independent of the base field.
SWARM AND EVOLUTIONARY COMPUTATION
(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
Engineering, Electrical & Electronic
Aleksandar Minja, Vojin Senk
Summary: This paper introduces the MAP decoding method for Kerdock and Preparata codes and also presents a sub-optimal APP decoder. Both these decoders show good error-correcting performance and complexity.
IEEE TRANSACTIONS ON COMMUNICATIONS
(2022)
Article
Computer Science, Theory & Methods
Ivan Chajda, Helmut Laenger
Summary: We prove that for orthomodular poset P = (P, <, ', 0, 1) of finite height, two operators can be defined to form an adjoint pair with respect to an order-like relation on the power set of P. This allows us to introduce the operator residuated poset corresponding to P, from which the original orthomodular poset can be recovered. We also demonstrate that this construction of operators can be applied to weakly orthomodular and dually weakly orthomodular posets. Examples of such posets are provided.
FUZZY SETS AND SYSTEMS
(2023)
Article
Engineering, Electrical & Electronic
Toshiki Matsumine, Toshiaki Koike-Akino, Hideki Ochiai
Summary: This paper proposes a new probabilistic amplitude shaping (PAS) scheme based on short linear block codes, which generates capacity-approaching signal distribution during the decoding process of linear block codes for a given information bit sequence. By choosing perfect binary codes, the proposed scheme offers a shaping gain of approximately 0.3-1.0 dB. Comparative analysis with enumerative sphere shaping (ESS) shows that the proposed scheme achieves significantly lower storage complexity and computational complexity at the receiver while maintaining comparable block error rate performance.
IEEE TRANSACTIONS ON COMMUNICATIONS
(2021)
Article
Computer Science, Theory & Methods
Ivan Chajda, Miroslav Kolarik, Helmut Laenger
Summary: It is shown that every poset P with an antitone involution can be extended to a commutative integral residuated poset E(P), and if P is a lattice, then E(P) is also a lattice.
FUZZY SETS AND SYSTEMS
(2021)
Article
Computer Science, Information Systems
Jose Gomez-Torrecillas, Gabriel Navarro, Jose Patricio Sanchez-Hernandez
Summary: A large class of MDS linear codes with efficient decoding algorithms were constructed in this paper, using methods from Linear Algebra. The codes were presented through parity-check matrices, and the decoding algorithm involved matrix and linear map manipulations. The proof of correctness of the decoding algorithm in a more sophisticated mathematical context was postponed to a later section. Additionally, the Reed-Solomon skew-differential codes were positioned within the general context of codes defined by skew polynomial rings.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2021)
Review
Automation & Control Systems
Cristiano Torezzan, Luciano Panek, Marcelo Firer
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2016)
Article
Mathematics
Luciano Panek, Marcelo Firer, Marcelo Muniz Silva Alves
DISCRETE MATHEMATICS
(2009)
Article
Computer Science, Information Systems
Luciano Panek, Marcelo Firer, Marcelo Muniz Silva Alves
IEEE TRANSACTIONS ON INFORMATION THEORY
(2010)
Article
Computer Science, Information Systems
Luciano Panek, Nayene Michele Paiao Panek
Summary: The paper investigates metrics on linear spaces induced by chain orders and weights, determines the cardinality and classification of optimal anticodes, and identifies all diameter perfect codes for a set of relevant instances on the metric spaces.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2021)
Article
Computer Science, Information Systems
Luciano Panek, Jerry Anderson Pinheiro
IEEE TRANSACTIONS ON INFORMATION THEORY
(2020)
Article
Computer Science, Theory & Methods
Luciano Panek, Jerry Anderson Pinheiro, Marcelo Muniz Alves, Marcelo Firer
ADVANCES IN MATHEMATICS OF COMMUNICATIONS
(2020)
Proceedings Paper
Physics, Applied
M. Firer, L. Panek, L. L. R. Rifo
XI BRAZILIAN MEETING ON BAYESIAN STATISTICS (EBEB 2012)
(2012)
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)
