Article
Computer Science, Theory & Methods
Cuiling Fan, An Wang, Li Xu
Summary: This paper presents a method for constructing several NMDS codes with different parameters by adding suitable projective points into known arcs in PG(2, q). These codes are linearly inequivalent to the NMDS codes constructed from elliptic curves, and their weight distributions are determined.
DESIGNS CODES AND CRYPTOGRAPHY
(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
Computer Science, Theory & Methods
Sascha Kurz, Ivan Landjev, Assia Rousseva
Summary: This paper presents an example of using extendability results and introduces the concept of (t mod q)-arcs as a framework for extendability results. By completing the partial classification, two exceptional examples are discovered and an extended proof is provided.
ADVANCES IN MATHEMATICS OF COMMUNICATIONS
(2023)
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
Computer Science, Information Systems
Jurgen Bierbrauer, Stefano Marcugini, Fernanda Pambianco
Summary: The study focuses on determining the optimal parameters of additive quaternary codes, particularly in the case of dimension k = 2.5, and presents new optimal solutions. Additionally, a novel proof method for the existence of binary linear codes is provided.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2021)
Article
Mathematics, Applied
Zeyu Ye, Qunying Liao
Summary: Cyclic codes, a subclass of linear codes, are widely used in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. In 2013, Ding et al. proposed nine open problems regarding optimal ternary cyclic codes. Some of these problems have been completely or partially solved. This paper focuses on the 7th problem and provides an incomplete answer and a counterexample by determining the root set of special polynomials over finite fields. Additionally, two new classes of optimal ternary cyclic codes are constructed based on studying special polynomials over finite fields.
FINITE FIELDS AND THEIR APPLICATIONS
(2023)
Article
Mathematics, Applied
Yuto Inoue, Tatsuya Maruta
Summary: In this paper, a new concept of geometric extending for linear codes over finite fields is introduced, and the extendability of divisible codes is considered. As an application, new Griesmer [n, 5, d](q) codes are constructed for specific ranges of d with q >= 3 by combining known geometric methods like projective dual, geometric extending, and geometric puncturing.
FINITE FIELDS AND THEIR APPLICATIONS
(2021)
Article
Mathematics
Toshiharu Sawashima, Tatsuya Maruta
Summary: The study proves the nonexistence of some ternary linear codes of dimension 6 and introduces proof techniques such as i-Max and i-Max-NS to rule out possible weights of codewords through projective geometry.
DISCRETE MATHEMATICS
(2021)
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
Elyassaf Loyfer, Nati Linial
Summary: We have developed a new family of linear programs that provide upper bounds on the rate of binary linear codes with a given distance. These bounds are specific to linear codes and our LPs yield increasingly tighter upper bounds as the control parameter increases. Numerical experiments have shown significant improvement over Delsarte's LP. We expect our work to lead to new improved asymptotic upper bounds on the rate of linear codes, considering the convincing numerical results and the variety of tools available for asymptotic analysis. Additionally, we have provided a new proof of convergence for closely related LPs previously introduced by Coregliano, Jeronimo, and Jones.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2023)
Article
Mathematics, Applied
Kelly Isham
Summary: This paper investigates the counting problem of n-arcs in projective space and determines a formula for such counting. Exact expressions for the number of n-arcs in P3(Fq) are provided within a specific range, which are polynomial/quasipolynomial in q depending on the value of n. The study is also extended to higher-dimensional projective space.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Computer Science, Theory & Methods
Sascha Kurz, Eitan Yaakobi
Summary: This work investigates private information retrieval (PIR) codes and analyzes the minimization of encoded bits under different parameters, with a specific focus on improving existing results.
DESIGNS CODES AND CRYPTOGRAPHY
(2021)
Article
Computer Science, Information Systems
Chunming Tang, Cunsheng Ding
Summary: This paper introduces the binary quadratic-residue codes and the punctured Reed-Muller codes R.2((ru-1)/2, m)), two families of binary cyclic codes with special parameters and minimum distance bounds. The objective of the paper is to construct two families of binary cyclic codes with length 2^m-1 and dimension near 2m-1.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Computer Science, Interdisciplinary Applications
Krzysztof Fleszar
Summary: VSBPPTW is a variant of the classical bin packing problem that minimizes the cost of using different sized bins to pack all items, while ensuring time-compatibility between the items in each bin. A new MILP model is proposed to address VSBPPTW, which fills each bin with items from a maximal time-compatible set and uses a compressed network flow formulation. New fast heuristics are also introduced to improve the solution quality.
COMPUTERS & INDUSTRIAL ENGINEERING
(2023)
Article
Computer Science, Theory & Methods
Toshiharu Sawashima, Tatsuya Maruta
Summary: One of the fundamental problems in coding theory is to find the minimum length n for a linear code of dimension k and minimum weight d. Using geometric methods and a new extension theorem, we determine n(3)(6, d) for some values of d.
ADVANCES IN MATHEMATICS OF COMMUNICATIONS
(2023)
Article
Mathematics, Applied
Daniele Bartoli, Matteo Bonini
Summary: This paper constructs a class of planar polynomials and classifies them using connections with algebraic curves over finite fields to determine whether they are planar or not.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Mathematics
Daniele Bartoli, Giacomo Micheli
Summary: This paper provides a complete solution to a classical problem in finite geometry, constructing the smallest complete m-arcs when q is relatively large. By developing a Galois theoretical machinery, the paper demonstrates the m-completeness of the arcs.
Article
Mathematics, Applied
Daniele Bartoli, Giovanni Zini, Ferdinando Zullo
Summary: This paper investigates the classification and properties of scattered polynomials over a finite field, introduces the concepts of L-qt-partially scattered and R-qt-partially scattered polynomials, and demonstrates that L-qt-partially scattered is the most difficult property to preserve. It also explores a large family of R-qt-partially scattered polynomials, introduces equivalence notions, and uses geometric arguments to determine equivalence classes under specific actions.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Mathematics, Applied
Daniele Bartoli, Antonio Cossidente, Giuseppe Marino, Francesco Pavese
Summary: This paper studies cutting blocking sets over finite fields and their applications in constructing linear codes. By using pairwise disjoint sets of lines, minimal linear codes with linear growth of length with respect to dimension can be constructed. Additionally, the authors propose two new constructions of cutting blocking sets, demonstrating that they are smaller than the known results.
FORUM MATHEMATICUM
(2022)
Article
Computer Science, Theory & Methods
Daniele Bartoli, Marco Calderini, Constanza Riera, Pantelimon Stanica
Summary: In this paper, we investigate the c-differential uniformity of piecewise defined functions and present improved results. We also examine concatenations of functions with low differential uniformity and provide proofs for several findings. These studies are crucial for understanding the differential uniformity of functions.
CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES
(2022)
Article
Mathematics, Applied
Daniele Bartoli, Matteo Bonini
Summary: This article presents an alternative proof of the necessary conditions on A, B, and their application in algebraic geometry over finite fields and fast symbolic computations.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2023)
Article
Mathematics, Applied
Daniele Bartoli, Giuseppe Marino, Alessandro Neri
Summary: This paper investigates minimal rank-metric codes, or linear cutting blocking sets, which are characterized in terms of the second generalized rank weight through their connection with evasiveness properties of the associated q-system. The authors provide the first construction of a family of Fq(m)-linear MRD codes of length 2m that are not obtained as a direct sum of two smaller MRD codes. The family exhibits better parameters and possess generalized rank weights strictly larger than those of previously known MRD codes, indicating that not all MRD codes have the same generalized rank weights, unlike in the Hamming metric setting.
ANNALI DI MATEMATICA PURA ED APPLICATA
(2023)
Article
Mathematics, Applied
Daniele Bartoli, Giovanni Zini, Ferdinando Zullo
Summary: We investigate the tensor rank of some Gabidulin codes of small dimension, and determine the tensor rank of any rank metric code equivalent to an 8-dimensional F-q-linear generalized Gabidulin code in F-q(4x4). This reveals the first infinite family of Gabidulin codes that are not minimum tensor rank.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Computer Science, Theory & Methods
Daniele Bartoli, Lins Denaux
Summary: This paper focuses on the codes arising from the incidence of points and hyperplanes in a projective space, with a particular interest in small weight codewords. The main result of this work shows that, under certain conditions, a codeword with weight smaller than a certain threshold can be expressed as a linear combination of hyperplanes. This result is further used to provide a graph-theoretical sufficient condition for determining whether these small weight codewords are minimal.
ADVANCES IN MATHEMATICS OF COMMUNICATIONS
(2023)
Editorial Material
Computer Science, Theory & Methods
Daniele Bartoli, Martino Borello, Anna-Lena Horlemann
ADVANCES IN MATHEMATICS OF COMMUNICATIONS
(2023)
Article
Mathematics, Applied
Daniele Bartoli, Marco Calderini, Marco Timpanella
Summary: This paper investigates the conditions for a polynomial to be a crooked function and introduces the concept of exceptional crooked functions. Using a connection with algebraic varieties over finite fields, non-existence results of exceptional crooked functions are provided.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Mathematics, Applied
Daniele Bartoli, Martino Borello
Summary: In recent years, there has been significant interest in strong blocking sets and their counterparts, minimal codes. By combining the concatenating construction of codes with a geometric understanding of minimality, we provide explicit infinite families of small strong blocking sets whose size scales linearly with the dimension of the ambient projective spaces. As a result, small saturating sets are also obtained.
SIAM JOURNAL ON DISCRETE MATHEMATICS
(2023)
Article
Mathematics, Applied
Daniele Bartoli, Marco Timpanella
Summary: This paper investigates the non-existence problem of rational perfect nonlinear functions over a finite field, using deep results about the number of points of algebraic varieties over finite fields.
ANNALI DI MATEMATICA PURA ED APPLICATA
(2023)
Article
Computer Science, Theory & Methods
Daniele Bartoli, Giuliana Fatabbi, Francesco Ghiandoni
Summary: We investigate APN functions represented as rational functions and provide non-existence results by connecting these functions to algebraic varieties over finite fields. This approach allows for classifying families of functions where previous approaches are not applicable.
DESIGNS CODES AND CRYPTOGRAPHY
(2023)
Article
Computer Science, Information Systems
Daniele Bartoli, Giovanni Zini, Ferdinando Zullo
Summary: In this paper, we propose a unified algebraic description of F(q)n-linear maximum rank distance (MRD) codes, and introduce the concept of exceptional linear MRD codes with a given index. This connection extends the notion of exceptionality for scattered polynomials in the rank metric framework and generalizes Moore sets in the monomial MRD context. We make progress towards the classification of exceptional linear MRD codes by proving that the codes of index zero are generalized Gabidulin codes, and showing that in the positive index case, the code contains an exceptional scattered polynomial of the same index.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2023)
