Article
Mathematics
Tomislav Guzvic
Summary: By studying all elliptic curves E defined over Q and all number fields K such that [K:Q] = pq, we determine all the possibilities for the torsion points E(K).
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2023)
Article
Computer Science, Information Systems
Dongchun Han, Cuiling Fan
Summary: This paper analyzes the NMDS properties of Roth-Lempel linear codes and obtains the necessary and sufficient condition for Roth-Lempel codes to be NMDS. The weight distributions of Roth-Lempel codes with length q + 2 and dimension 3 = k = q are completely determined. In addition, by analyzing the upper bound for the code lengths of elliptic curve MDS codes, the linear inequivalence between Roth-Lempel NMDS codes and elliptic curve NMDS codes is illustrated when their corresponding code lengths exceed 4(q + 2vq + 1)/5 + 1.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2023)
Article
Computer Science, Information Systems
Dongchun Han, Yuan Ren
Summary: This paper aims to derive an upper bound for the maximal length of MDS elliptic codes over F-q with dimension 3 <= k <= q+1-2 root q / 10. The result improves an earlier bound and confirms a conjecture. Notably, the proposed upper bound is tight for odd dimensions and can be achieved by well-designed MDS elliptic codes.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2023)
Article
Mathematics
Yuan Ren, Dongchun Han
Summary: This paper generalizes the construction of algebraic-geometric codes to the setting of F-q-linear Fqt-codes and shows that codes arising from the projective line yield MDS (resp. self-dual MDS) F-q-linear F-qt-codes (resp. when q is a power of 2). Additionally, a tight upper bound for the maximal length of primitive MDS elliptic F-q-linear F-qt-codes with F-q-dimension k divided by t and satisfying 3 ≤ k/t ≤ q+1-2 root q/20 is derived.
DISCRETE MATHEMATICS
(2023)
Article
Computer Science, Information Systems
Shu Liu, Liming Ma, Ting-Yi Wu, Chaoping Xing
Summary: Constructing codes with good parameters is a fundamental problem in coding theory. This paper introduces a new explicit construction method for (q + 1)-ary nonlinear codes using algebraic function fields. The codes constructed in this way have better parameters compared to other codes.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2023)
Article
Mathematics, Applied
Boran Kim
Summary: In this study, we focus on locally recoverable codes in Hermitian function fields and provide an explicit formula for the improved lower bound of minimum distance. We also present a criterion for suggesting an improved lower bound based on divisors with certain places.
Article
Computer Science, Information Systems
Peter Beelen, Johan Rosenkilde, Grigory Solomatov
Summary: This paper presents an efficient list decoding algorithm for algebraic geometry codes in the style of Guruswami-Sudan. The algorithm can decode any such code using operations in the underlying finite field, taking into account various parameters. The interpolation step utilizes known algorithms for univariate polynomial matrices, while the root-finding step is solved using existing algorithms for root-finding over univariate power series.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Computer Science, Theory & Methods
Daniele Bartoli, Maria Montanucci, Giovanni Zini
Summary: This paper provides a construction of quantum codes from self-orthogonal algebraic geometry codes based on CSS construction and peculiar properties of Swiss curves. The construction extends a previous method and applies to well-known algebraic curves such as Castle curves and GK curves that are shown to be Swiss curves.
DESIGNS CODES AND CRYPTOGRAPHY
(2021)
Article
Engineering, Electrical & Electronic
Yunqi Wan, Li Chen, Fangguo Zhang
Summary: This paper proposes an algebraic soft decoding (ASD) method for one-point elliptic codes, which solves the interpolation problem by reducing the module basis. The interpolation polynomial Q(x, y, z) is a minimum candidate of a Grobner basis in ASD. By defining an interpolation ideal based on a multiplicity matrix, an equivalent interpolation module can be obtained. Furthermore, the paper introduces the re-encoding transform (ReT) to simplify the basis reduction complexity. The effectiveness of this technique is shown through numerical results and compared to the conventional Kotter’s interpolation.
IEEE TRANSACTIONS ON COMMUNICATIONS
(2022)
Article
Mathematics
Fan Peng, Hao Chen, Chang-An Zhao
Summary: Algebraic geometric secret sharing schemes are proposed for establishing the fundamental theorem in information-theoretically secure multiparty computation. The schemes demonstrate quasithreshold properties and are analyzed in terms of subsets of players' ability to recover the secret or have no knowledge of it. The study explores the asymptotic threshold behavior of these schemes over large finite fields and when the genus approaches infinity with a fixed base field size.
PACIFIC JOURNAL OF MATHEMATICS
(2022)
Article
Telecommunications
Yunqi Wan, Jiongyue Xing
Summary: This paper proposes a low-complexity Kotter's interpolation algorithm for list decoding of elliptic codes, which significantly reduces the interpolation complexity by arranging interpolation constraints and utilizing weighted degree constraints. The complexity reduction of the proposed interpolation technique is analyzed and validated through simulation results.
IEEE COMMUNICATIONS LETTERS
(2021)
Article
Mathematics, Applied
Jigu Kim, Yoonjin Lee
Summary: This paper provides lower bounds on the caliber numbers of parametric real quadratic fields and Richaud-Degert type real quadratic fields. The limits of congruent numbers and algebraic ranks are investigated by varying different parameters.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics
Adel Alahmadi, Patrick Sole, Ramy Taki Eldin
Summary: In this paper, a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields is introduced, which is analogous to the well-known BCH bound for cyclic codes. Some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes are proposed based on this BCH-like bound. A novel bound that takes into account the Chinese remainder theorem approach to QT codes as well as the BCH bound of constacyclic codes is also proposed, which does not require a specific form of code generator or calculations in any extension field.
Article
Mathematics, Applied
Gustavo Cabana, Maria Chara, Ricardo Podesta, Ricardo Toledano
Summary: In this paper, we introduce the study of cyclic algebraic geometry codes and provide conditions for constructing them in the context of algebraic function fields over a finite field using their automorphism group. We prove the close relationship between cyclic algebraic geometry codes constructed in this way and cyclic extensions. Furthermore, we conduct a detailed study on the monomial equivalence of cyclic algebraic geometry codes constructed with our method in the case of a rational function field.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Mathematics
Jin Cao, Hossein Movasati, Roberto Villaflor Loyola
Summary: This article discusses the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space provides a natural algebro-geometric framework for higher genus quasi-Jacobi forms of index zero and their differential equations represented as vector fields. Explicit computations of the Gauss-Manin connection and such vector fields are presented for the case of elliptic curves.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(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)
