Article
Computer Science, Information Systems
Edmundo J. Huertas, Alberto Lastra, Anier Soria-Lorente
Summary: In this article, for the first time, the non-standard properties of monic polynomials are applied in a watermarking problem, revealing differences compared to the standard case.
Article
Physics, Mathematical
Nicolas Crampe, Julien Gaboriaud, Loic Poulain d'Andecy, Luc Vinet
Summary: We compute the matrix elements of SO(3) in any finite-dimensional irreducible representation of sl3. They are expressed in terms of a double sum of products of Krawtchouk and Racah polynomials which generalize the Griffiths-Krawtchouk polynomials. The recurrence and difference relations of these polynomials are obtained. The proof is based on the decomposition of a general three-dimensional rotation and the embedding of sl(2) in sl(3). The appearance of Racah polynomials is explained algebraically by the relations of the two sl(2) Casimir elements. The centralizer in U(sl(3)) of the Cartan subalgebra is shown to be generated by these elements.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Mathematics
Albertus C. den Brinker
Summary: The study proposes an algorithm for stable determination of orthogonal basis for Krawtchouk functions, achieved by defining proper initial points, balancing the order of recursion execution, and adaptively restricting the range of equation application. The adaptation is controlled by user-specified deviation from unit norm, with theoretical background provided, algorithmic concept explained, and the effect of controlled accuracy demonstrated through examples.
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
Chemistry, Analytical
Sadiq H. Abdulhussain, Basheera M. Mahmmod, Marwah Abdulrazzaq Naser, Muntadher Qasim Alsabah, Roslizah Ali, S. A. R. Al-Haddad
Summary: A new scheme for handwritten numeral recognition using Hybrid orthogonal polynomials is proposed in this study, where gradient and smoothed features are extracted and support vector machine is used for classification. Experimental results show that the proposed method achieves almost the highest recognition accuracy in different datasets and is robust against noise distortion.
Article
Computer Science, Theory & Methods
Kanat Abdukhalikov, Duy Ho
Summary: By utilizing characterizations of ovals, KM-arcs, and elliptic quadrics described in polar coordinates, we generate several families of linear codes with properties such as LCD, self-orthogonality, three-weight, and four-weight. We also showcase their applications in quantum codes.
DESIGNS CODES AND CRYPTOGRAPHY
(2023)
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
Rupert Li
Summary: The Delsarte linear program is used to bound the size of codes by taking a linear relaxation from codes to quasicodes. We study the unique optimum values of this linear program for different (n, d) pairs, proving that it has a unique optimum when d > n/2 or d <= 2. Using the Krawtchouk decomposition of a quasicode, we show that optima to certain linear programs have identical Krawtchouk decompositions, revealing a parity phenomenon in the Delsarte linear programs. We generalize the concept of extending and puncturing codes to quasicodes and characterize these pairs of optima, demonstrating a symmetry property that halves the number of decision variables.
DESIGNS CODES AND CRYPTOGRAPHY
(2023)
Article
Mathematics, Applied
Hiroshi Miki, Satoshi Tsujimoto, Luc Vinet
Summary: This study investigates the Darboux transformations of Krawtchouk polynomials and considers all possible exceptional Krawtchouk polynomials obtainable from a single-step Darboux transformation. The properties of these exceptional Krawtchouk polynomials, including the Diophantine ones and the recurrence relations, are obtained.
JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS
(2023)
Article
Computer Science, Theory & Methods
Dev Akre, Nuh Aydin, Matthew Harrington, Saurav Pandey
Summary: This study generalized an algorithm for testing equivalence of cyclic codes to constacyclic codes and discovered 22 new constacyclic codes that improve the minimum distances of best known linear codes.
DESIGNS CODES AND CRYPTOGRAPHY
(2023)
Article
Mathematics, Applied
Shitao Li, Minjia Shi
Summary: In this paper, we construct two infinite families of new two-weight codes over Z(2m) by their generator matrices, which generalize the previous results. We also construct some optimal codes and prove that all codes in one of the families are self-orthogonal. Finally, we determine the linearity of the Gray images of the codes constructed for Lee metric.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Mathematics, Applied
John Vincent S. Morales
Summary: This paper investigates a Lie algebra constructed by generators and relations, and its isomorphism with the three-point sl(2) loop algebra. By studying the subalgebras within the Lie algebra and their relationship with direct sums, some conclusions about modules are obtained, which are also related to Krawtchouk polynomials.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Computer Science, Theory & Methods
Stefka Bouyuklieva
Summary: Linear complementary dual codes, known as LCD codes, have trivial intersections with their dual codes and are widely used in data storage and cryptography. This paper proves properties of binary LCD codes using shortened and punctured codes, as well as presents inequalities for the largest minimum weight of binary LCD [n, k] codes. Additionally, tables with values and classification results are provided for reference.
DESIGNS CODES AND CRYPTOGRAPHY
(2021)
Article
Mathematics, Applied
Minjia Shi, Ferruh Ozbudak, Li Xu, Patrick Sole
Summary: Double Toeplitz (DT) codes are introduced with a focus on the form of their generator matrix and conditions for being LCD codes. The explicit determination of the spectrum of a tridiagonal symmetric matrix using Dickson polynomials is discussed, along with the construction of optimal or quasi-optimal examples of binary and ternary LCD codes through a special concatenation process.
FINITE FIELDS AND THEIR APPLICATIONS
(2021)
Article
Computer Science, Theory & Methods
Hyun Seung Choi, Boran Kim
Summary: In this paper, the authors investigate cyclic codes over the ring Rp = Fp+uFp+ vFp, where u2 = v2 = uv = vu = 0 (p: a prime number). They derive the generators of ideals of Rp[x]/(xn - 1) for cyclic codes over Rp of arbitrary length n. Moreover, explicit generators are obtained for cyclic codes, their duals, self-orthogonal codes, and self-dual codes over Rp of length n when gcd(n, p) = 1. The authors also present a Gray map that shows the image of a cyclic code over this map is quasi-cyclic, and determine the index of this image. The mass formulae of cyclic self-orthogonal and LCD codes over Rp of length n are derived as well.
ADVANCES IN MATHEMATICS OF COMMUNICATIONS
(2023)
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, 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
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
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
Mathematics, Applied
Chunming Tang, Peng Han, Qi Wang, Jun Zhang, Yanfeng Qi
Summary: In this paper, we study the binomial Boolean functions and prove that fa,b is a bent function if certain conditions are met. The proof involves computing Walsh coefficients, Gauss sums, and graph theory. This resolves an open problem posed by Mesnager.
FINITE FIELDS AND THEIR APPLICATIONS
(2023)
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
Mathematics, Applied
Can Xiang, Chunming Tang
Summary: This paper investigates extended codes of a class of binary cyclic codes and their dual codes, showing that these codes possess 3-designs. Additionally, the paper obtains some shortened codes from the studied cyclic codes and determines their parameters explicitly. Some of the shortened codes are found to be either optimal or almost optimal.
FINITE FIELDS AND THEIR APPLICATIONS
(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
Yue Zhao, Chunming Tang, Yanfeng Qi
Summary: The importance of interactions between groups, linear codes, and t-designs has been recognized for decades. Linear codes invariant under group actions on code coordinates have important applications in constructing combinatorial t-designs. This paper presents two families of cyclic codes that are invariant under the action of PGL(2, q) and support 3-designs. The parameters of the codes in the first family are [q + 1, 4, q - 5](q), while the parameters of the codes in the second family are [q + 1, q - 3, 4](q). The paper also points out the invariance of the support sets of codewords under Stab(Uq+1) and the corresponding support of 3-designs.
ARITHMETIC OF FINITE FIELDS, WAIFI 2022
(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)
