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Matrix graphs and mrd codes over finite principal ideal rings

FINITE FIELDS AND THEIR APPLICATIONS 2020年 66卷 101705-101705页

作者： Sirisuk, Siripong Meemark, Yotsanan Chulalongkorn Univ Dept Math & Comp Sci Fac Sci Bangkok 10330 Thailand Thammasat Univ Fac Sci & Technol Dept Math & Stat Pathum Thani 12120 Thailand

Let R be a finite principal ideal ring and m, n, d positive integers. In this paper, we study the matrix graph over R which is the graph whose vertices are m x n matrices over Rand two matrices A and Bare adjacent if and only if 0 < rank(A-B) < d. We show that this graph is a connected vertex transitive graph. The distance, diameter, independence number, clique number and chromatic number of this graph are also determined. This graph can be applied to study mrd codes over R. We obtain that a maximal independent set of the matrix graph is a maximum rank distance (mrd) code and vice versa. Moreover, we show the existence of linear mrd codes over R. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Matrix graph mrd code Finite principal ideal ring

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Rank-metric codes, linear sets, and their duality

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DESIGNS codeS AND CRYPTOGRAPHY 2020年第4期88卷 655-675页

作者： Sheekey, John Van de Voorde, Geertrui Univ Coll Dublin Dublin Ireland Univ Canterbury Private Bag 4800 Christchurch 8140 New Zealand

In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of Sheekey (Adv Math Commun 10:475-488, 2016, Sect. 5) on linear sets on a projective line. We extend this to linear sets in arbitrary dimension, giving the connection between two constructions for linear sets defined in Lunardon (J Comb Theory Ser A 149:1-20, 2017). Finally, we then exploit this connection by using the MacWilliams identities to obtain information about the possible weight distribution of a linear set of rank n on a projective line PG(1,qn)

关键词： mrd code Weight distribution Linear set Scattered with respect to hyperplanes

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Partition-balanced families of codes and asymptotic enumeration in coding theory

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JOURNAL OF COMBINATORIAL THEORY SERIES A 2020年 171卷

作者： Byrne, Eimear Ravagnani, Alberto Univ Coll Dublin Sch Math & Stat Belfield Ireland

We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise asymptotic estimates on the density functions of several classes of codes that are extremal with respect to minimum distance, covering radius, and maximality. The techniques developed in this paper apply to various distance functions, including the Hamming and the rank metric distances. Applications of our results show that, unlike the F-qm-linear mrd codes, the F-q-linear mrd codes are not dense in the family of codes of the same dimension. More precisely, we show that the density of F-q-linear mrd codes in F-q(nxm) in the set of all matrix codes of the same dimension is asymptotically at most 1/2, both as q -> +infinity and as m -> +infinity. We also prove that MDS and F-qm-linear mrd codes are dense in the family of maximal codes. Although there does not exist a direct analogue of the redundancy bound for the covering radius of Fq-linear rank metric codes, we show that a similar bound is satisfied by a uniformly random matrix code with high probability. In particular, we prove that codes meeting this bound are dense. Finally, we compute the average weight distribution of linear codes in the rank metric, and other parameters that generalize the total weight of a linear code. (C) 2019 Published by Elsevier Inc.

关键词： Asymptotic enumeration Partition-balanced family Error-correcting code Density function Hamming metric Rank metric MDS code mrd code

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Rank-metric codes andq-polymatroids

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JOURNAL OF ALGEBRAIC COMBINATORICS 2020年第1期52卷 1-19页

作者： Gorla, Elisa Jurrius, Relinde Lopez, Hiram H. Ravagnani, Alberto Univ Neuchatel Inst Matemat Neuchatel Switzerland Netherlands Def Acad Fac Mil Sci Den Helder Netherlands Cleveland State Univ Dept Math Cleveland OH 44115 USA Univ Coll Dublin Sch Math & Stat Dublin Ireland

This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduceq-polymatroids, theq-analogue of polymatroids, and develop their basic properties. We associate a pair ofq-polymatroids with a rank-metric code and show that several invariants and structural properties of the code, such as generalized weights, the property of being mrd or an optimal anticode, and duality, are captured by the associated combinatorial object.

关键词： Rank-metric code Generalized weights q-polymatroid mrd code Optimal anticode Duality

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Generalized bilinear forms graphs and mrd codes over a residue class ring

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FINITE FIELDS AND THEIR APPLICATIONS 2018年 51卷 306-324页

作者： Huang, Li-Ping Changsha Univ Sci & Technol Sch Math & Statis Changsha 410004 Hunan Peoples R China

We investigate the generalized bilinear forms graph Gamma(d) over a residue class ring Z(ps). The graph Gamma(d) is a connected vertex-transitive graph, and we completely determine its independence number, clique number, chromatic number and maximum cliques. We also prove that cores of both Gamma(d) and its complement are maximum cliques. The graph Gamma(d) is useful for error-correcting codes. We show that there is a largest independent set of Gamma(d) which is a linear mrd code over Z(ps). (C) 2018 Elsevier Inc. All rights reserved.

关键词： Bilinear forms graph Residue class ring Independence number mrd code Core Maximum clique

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On the classification of exceptional scattered polynomials

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JOURNAL OF COMBINATORIAL THEORY SERIES A 2021年 179卷 105386-105386页

作者： Bartoli, Daniele Montanucci, Maria Univ Perugia Dipartimento Matemat & Informat Perugia Italy Tech Univ Denmark Dept Appl Math & Comp Sci Asmussens AlleBldg 303B DK-2800 Lyngby Denmark

Let f(X) is an element of F-qr[X] be a q-polynomial. If the F-q-subspace U = {(x(qt), f(x)) vertical bar x is an element of F-qn} defines a maximum scattered linear set, then we call f(X) a scattered polynomial of index t. The asymptotic behavior of scattered polynomials of index t is an interesting open problem. In this sense, exceptional scattered polynomials of index tare those for which U is a maximum scattered linear set in PG(1, q(mr)) for infinitely many m. The classifications of exceptional scattered monic polynomials of index 0(for q > 5) and of index 1 were obtained in [1]. In this paper we complete the classifications of exceptional scattered monic polynomials of index 0 for q <= 4. Also, some partial classifications are obtained for arbitrary t. As a consequence, the classification of exceptional scattered monic polynomials of index 2 is given. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Maximum scattered linear set mrd code Algebraic curve Hasse-Weil bound

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On certain linearized polynomials with high degree and kernel of small dimension

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JOURNAL OF PURE AND APPLIED ALGEBRA 2021年第2期225卷 106491-106491页

作者： Polverino, Olga Zini, Giovanni Zullo, Ferdinando Univ Campania Luigi Vanvitelli Dipartimento Matemat & Fis Viale Lincoln 5 I-81100 Caserta Italy

Let f be the F-q-linear map over F-q(2n) defined by x bar right arrow x ax(q)(s) + bx(qn+s) with gcd(n, s) = 1. It is known that the kernel of f has dimension at most 2, as proved by Csajbok et al. in [9]. For n big enough, e.g. n >= 5 when s = 1, we classify the values of b/a such that the kernel of f has dimension at most 1. To this aim, we translate the problem into the study of some algebraic curves of small degree with respect to the degree of f;this allows to use intersection theory and function field theory together with the Hasse-Weil bound. Our result implies a non-scatteredness result for certain high degree scattered binomials, and the asymptotic classification of a family of rank metric codes. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Linearized polynomial Algebraic curve Linear set mrd code Hasse-Weil bound

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Tensor Representation of Rank-Metric codes

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SIAM JOURNAL ON APPLIED ALGEBRA AND GEOMETRY 2019年第4期3卷 614-643页

作者： Byrne, Eimear Neri, Alessandro Ravagnani, Alberto Sheekey, John Univ Coll Dublin Sch Math & Stat Dublin 4 Ireland Univ Zurich Inst Math CH-8057 Zurich Switzerland

We present the theory of rank-metric codes with respect to the 3-tensors that generate them. We define the generator tensor and the parity check tensor of a matrix code, and describe the properties of a code through these objects. We define the tensor rank of a code to be the tensor rank of its generating tensors, and propose that this quantity is a significant coding theoretic parameter. By a result on the tensor rank of Kruskal from the 1970s, the tensor rank of a rank-metric code of dimension k and minimum rank distance d is at least k+d-1 We call codes that meet this bound minimal tensor rank (MTR) codes. It is known from results in algebraic complexity theory that an MTR code implies the existence of a maximum distance separable (MDS) code. In this paper, we also address the converse problem, that of the existence of an MTR code, given an MDS code. We identify several parameters for which the converse holds and give explicit constructions of MTR codes using MDS codes. We furthermore define generalized tensor ranks, which give a refinement of the tensor rank as a code invariant. Moreover, we use these to distinguish inequivalent rank-metric codes.

关键词： rank metric tensor tensor rank tensor extremal code minimal tensor rank MTR code mrd code Delsarte-Gabidulin code generator tensor

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The State of Lexicodes and Ferrers Diagram Rank-Metric codes

The State of Lexicodes and Ferrers Diagram Rank-Metric Codes

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作者： Jared E. Antrobus University of Kentucky

学位级别：博士

In coding theory we wish to find as many codewords as possible, while simultane- ously maintaining high distance between codewords to ease the detection and correc- tion of errors. For linear codes, this translates to finding high-dimensional subspaces of a given metric space, where the induced distance between vectors stays above a specified minimum. In this work I describe the recent advances of this problem in the contexts of lexicodes and Ferrers diagram rank-metric codes. In the first chapter, we study lexicodes. For a ring R, we describe a lexicographic ordering of the left R-module R n. With this ordering we set up a greedy algorithm which sequentially selects vectors for which all linear combinations satisfy a given property. The resulting output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. We describe a generalization of the algorithm to finite principal ideal rings. In the second chapter, we investigate Ferrers diagram rank-metric codes, which play a role in the construction of subspace codes. A well-known upper bound for dimension of these codes is conjectured to be sharp. We describe several solved cases of the conjecture, and further contribute new ones. In addition, probabilities for maximal Ferrers diagram codes and mrd codes are investigated in a new light. It is shown that for growing field size, the limiting probability depends highly on the Ferrers diagram.

关键词： coding theory lexicode Ferrers diagram rank-metric mrd code

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On the number of inequivalent Gabidulin codes

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DESIGNS codeS AND CRYPTOGRAPHY 2018年第9期86卷 1973-1982页

作者： Schmidt, Kai-Uwe Zhou, Yue Paderborn Univ Dept Math D-33098 Paderborn Germany Natl Univ Def Technol Coll Sci Changsha 410073 Hunan Peoples R China

Maximum rank-distance (mrd) codes are extremal codes in the space of matrices over a finite field, equipped with the rank metric. Up to generalizations, the classical examples of such codes were constructed in the 1970s and are today known as Gabidulin codes. Motivated by several recent approaches to construct mrd codes that are inequivalent to Gabidulin codes, we study the equivalence issue for Gabidulin codes themselves. This shows in particular that the family of Gabidulin codes already contains a huge subset of mrd codes that are pairwise inequivalent, provided that n - 2 .

关键词： Rank metric code mrd code Finite field Semifield

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