Article
Mathematics
Joan-Josep Climent, Diego Napp, Raquel Pinto, Veronica Requena
Summary: This paper examines product convolutional codes represented by state-space models and explores how to derive these representations from horizontal and vertical convolutional codes. A systematic procedure for constructing such representations with minimal dimension is presented.
Article
Mathematics, Applied
Xiangdong Cheng
Summary: In this paper, the algebraic structure of (F2F8s)-F-r-additive codes is investigated. Generator polynomials for additive cyclic codes over F-8 and F2F8 are given. The properties of a linear map W : F2F8 -> F-2 are studied. The duals of additive cyclic codes over F2F8 are also explored, and it is found that the duals of any additive cyclic codes over F2F8 are also additive cyclic codes. Additionally, separable F2F8-additive cyclic codes are investigated.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Computer Science, Information Systems
Steven T. Dougherty, Josep Rifa, Merce Villanueva
Summary: This study establishes bounds on the rank and dimension of the kernel of additive generalised Hadamard (GH) codes and constructs additive GH codes for specific ranks and dimensions within these bounds. It is shown that the given bounds are tight for the case e = 2, allowing for the construction of additive GH codes for all allowable ranks and dimensions of the kernel between these bounds. Additionally, it is proven that these codes are self-orthogonal with respect to the trace Hermitian inner product and generate pure quantum codes.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2021)
Article
Computer Science, Theory & Methods
Whan Hyuk Choi, Jon Lark Kim
Summary: This paper introduces new methods for constructing symmetric self-dual codes and successfully generates several new self-dual codes with improved parameters and minimum distances, advancing the research in this field.
DESIGNS CODES AND CRYPTOGRAPHY
(2022)
Article
Chemistry, Analytical
Whai-En Chen, Yi-Bing Lin, Tai-Hsiang Yen, Syuan-Ru Peng, Yun-Wei Lin
Summary: This paper discusses the challenges of developing distributed intelligent systems in both the network and device domains. It proposes a low-code or no-code approach to automate code generation and introduces DeviceTalk, an environment that automatically generates code for IoT devices to speed up software development.
Article
Computer Science, Theory & Methods
Feng-qing Zhu, Xue-ping Wang
Summary: In this paper, the homogeneity of overlap functions is investigated using multiplicative generator pairs and additive generator pairs, respectively.
FUZZY SETS AND SYSTEMS
(2023)
Article
Computer Science, Theory & Methods
Hamidreza Eyvazi, Karim Samei, Batoul Savari
Summary: This paper investigates the properties of additive codes and Gray codes, and examines the linearity of linear codes. By introducing a new set S, it is possible to check the linearity of Gray codes more efficiently, reducing computational complexity. In addition, conditions for the linearity of cyclic codes and quadratic residue codes under Gray codes are presented.
DESIGNS CODES AND CRYPTOGRAPHY
(2022)
Article
Quantum Science & Technology
Om Prakash, Ram Krishna Verma, Ashutosh Singh
Summary: This paper constructs quantum and linear complementary dual (LCD) codes from skew constacyclic codes over the ring R. The explicit structure of skew constacyclic codes and their Euclidean as well as Hermitian duals over R are discussed. A necessary and sufficient condition for these codes to contain their Euclidean (Hermitian) duals is established. By applying CSS (Hermitian) construction, many new quantum codes with better parameters are obtained. Moreover, a necessary and sufficient condition is established for these codes over R to be Euclidean (Hermitian) LCD. Finally, many examples of MDS codes over Fpe are provided under the gray images of the skew Euclidean LCD codes.
QUANTUM INFORMATION PROCESSING
(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
Materials Science, Multidisciplinary
Junfeng Sun, Sajjan Parajuli, Kiran Shrestha, Jinhwa Park, Sagar Shrestha, Younsu Jung, Hyejin Park, Gyan Raj Koirala, Nadra Nasir, Seongryeong Kim, Han Truong, Haesook Jang, Jinkwan Lee, Jongchan Lee, Gyoujin Cho
Summary: Counterfeiting has led to significant global economic losses, highlighting the need for a cost-effective anti-counterfeiting platform. A wireless platform integrating various technologies has been developed to enable quick response code labels for anti-counterfeiting purposes.
ADVANCED MATERIALS TECHNOLOGIES
(2022)
Article
Mathematics, Applied
Ramy Taki Eldin
Summary: Galois duals of Multi-twisted (MT) codes are investigated in this study. The study describes a MT code C as a module over a principal ideal domain and proves a generator polynomial matrix formula for the Euclidean dual C' using the Hermite normal form of the GPM. The study distinguishes between the right and left Galois duals of a MT code, shows their properties, and establishes interconnected identities for them. The study also introduces the two-sided Galois dual and explores its conditions and properties.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Mathematics, Applied
Yingyu Luo, Yu Wang, Junjie Gu, Huihui Wang
Summary: This paper describes Jordan matrix algebras over a field using generators and relations, and proves that the minimum number of generators for some special Jordan matrix algebras over a field is 2.
Article
Computer Science, Theory & Methods
Minjia Shi, Na Liu, Jon-Lark Kim, Patrick Sole
Summary: A linear code is considered a linear complementary dual (LCD) code if it meets its dual trivially. Due to its application in the security of embarked electronics, LCD codes have gained attention recently. Additive codes over F-4, which are stable under codeword addition but not necessarily under scalar multiplication, can be considered additive complementary dual (ACD) codes if they meet their dual trivially. This research aims to study codes that meet their dual trivially. The techniques and problems used in studying LCD codes are potentially relevant to ACD codes. Interesting constructions of ACD codes, using the trace Hermitian and trace Euclidean inner product, are given with respect to binary codes.
DESIGNS CODES AND CRYPTOGRAPHY
(2023)
Article
Computer Science, Theory & Methods
Minjia Shi, Na Liu, Jon-Lark Kim, Patrick Sole
Summary: The study of Linear Complementary Dual (LCD) codes and Additive Complementary Dual (ACD) codes in the context of information security is a hot research topic.
DESIGNS CODES AND CRYPTOGRAPHY
(2022)
Article
Mathematics, Applied
Fatma Caliskan, Refia Aksoy
Summary: In this study, cyclic codes over the commutative principal ideal ring F2 X (F2 + vF2) with v2 = v are defined and some results on cyclic codes over F2 X (F2 + vF2) are obtained. The dual of a cyclic code over F2 X (F2 + vF2) depending on two inner products is also investigated. A generator polynomial of cyclic codes and the calculation of the number of cyclic codes over F2 X (F2 + vF2) are determined. Binary quasi-cyclic codes of length 3n and index 3 are shown to be the Gray images of cyclic codes over F2 X (F2 + vF2) of length n. Binary quantum error-correcting codes (QECCs) can be obtained from cyclic codes over F2 X (F2 + vF2).
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)
