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Common Complements of Linear Subspaces and the Sparseness of mrd codes\ast

SIAM JOURNAL ON APPLIED ALGEBRA AND GEOMETRY 2022年第2期6卷 79-110页

作者： Gruica, Anina Ravagnani, Alberto Eindhoven Univ Technol Dept Math & Comp Sci NL-5612 AZ Eindhoven Netherlands

Motivated by applications to the theory of rank-metric codes, we study the problem of estimating the number of common complements of a family of subspaces over a finite field in terms of the cardinality of the family and its intersection structure. We derive upper and lower bounds for this number, along with their asymptotic versions as the field size tends to infinity. We then use these bounds to describe the general behavior of common complements with respect to sparseness and density, showing that the decisive property is whether or not the number of spaces to be complemented is negligible with respect to the field size. By specializing our results to matrix spaces, we obtain upper and lower bounds for the number of maximum-rank-distance (mrd) codes in the rank metric. In particular, we answer an open question in coding theory, proving that mrd codes are sparse for all parameter sets as the field size grows, with only very few exceptions. We also investigate the density of mrd codes as their number of columns tends to infinity, obtaining a new asymptotic bound. Using properties of the Euler function from number theory, we then show that our bound improves on known results for most parameter sets. We conclude the paper by establishing general structural properties of the density function of rank-metric codes.

关键词： rank-metric code mrd code density

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A standard form for scattered linearized polynomials and properties of the related translation planes

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JOURNAL OF ALGEBRAIC COMBINATORICS 2024年第4期59卷 917-937页

作者： Longobardi, Giovanni Zanella, Corrado Univ Napoli Federico II Dipartimento Matemat & Applicaz Renato Caccioppol Via Cintia Monte S Angelo 1 I-80126 Naples Italy Univ Padua Dipartimento Tecn & Gest Sistemi Ind Stradella S Nicola 3 I-36100 Vicenza VI Italy

In this paper, we present results concerning the stabilizer Gf in GL(2,qn)of the subspace Uf={(x,f(x)):x is an element of Fqn},f(x)a scattered linearized polynomial in Fq n[x]. Each Gf contains the q-1 maps(x,y)->(ax,ay),a is an element of F & lowast;q. By virtue of the results of Beard (Duke Math J, 39:313-321, 1972) and Willett (Duke Math J40(3):701-704, 1973), the matrices in Gf are simultaneously diagonalizable. This has several consequences:(i)the polynomials such that |Gf|>q-1 have a standard form of type & sum;n/t-1j=0ajxqs+jtfor some s and t such that(s,t)=1,t>1 a divisor of n;(ii)this standard form is essentially unique;(iii)for n>2 andq>3, the translation plan eAf associated with f(x)admits nontrivial affine homologies if and only if|Gf|>q-1, and in that case those with axis through the origin form two groups of cardinality(qt-1)/(q-1)that exchange axes and coaxes;(iv)no plane of type Af,f(x)a scattered polynomial not of pseudoregulus type, is a generalized Andr & eacute;plane

关键词： Andre plane Linear set Linearized polynomial mrd code Partial spread Projective line Rank distance code Translation plane

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RANK-METRIC codeS, SEMIFIELDS, AND THE AVERAGE CRITICAL PROBLEM

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SIAM JOURNAL ON DISCRETE MATHEMATICS 2023年第2期37卷 1079-1117页

作者： Gruica, A. N. I. N. A. Ravagnani, A. L. B. E. R. T. O. Sheekey, J. O. H. N. Zullo, F. E. R. D. I. N. A. N. D. O. Eindhoven Univ Technol Dept Math & Comp Sci NL-5600 MB Eindhoven Netherlands Univ Coll Dublin Sch Math & Stat Dublin Ireland Univ Campania Luigi Vanvitelli Dipartimento Matemat & Fis I-81100 Caserta Italy

We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of mrd codes in the rank metric, and the Critical Problem for combinatorial geometries by Crapo and Rota. In the first part of the paper, we use methods from semifield theory to derive two lower bounds for the density function of full-rank, square mrd codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new look at the Critical Problem for combinatorial geometries, approaching it from a qualitative, often asymptotic, viewpoint. We illustrate the connection between this very classical problem and that of computing the asymptotic density of mrd codes. Finally, in the third part of the paper we study the asymptotic density of some special families of codes in the rank metric, including the symmetric, alternating, and Hermitian ones. In particular, we show that the optimal codes in these three contexts are sparse.

关键词： rank-metric code semifield critical problem asymptotic density enumerative com-binatorics density mrd code

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Linear Maximum Rank Distance codes of Exceptional Type

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IEEE TRANSACTIONS ON INFORMATION THEORY 2023年第6期69卷 3627-3636页

作者： Bartoli, Daniele Zini, Giovanni Zullo, Ferdinando Univ Perugia Dipartimento Matemat & Informat I-06123 Perugia Italy Univ Modena & Reggia Emilia Dipartimento Sci Fis Informat & Matematiche I-41121 Modena Italy Univ Campania Luigi Vanvitelli Dipartimento Matemat & Fis I-81100 Caserta Italy

Scattered polynomials of a given index over finite fields are intriguing rare objects with many connections within mathematics. Of particular interest are the exceptional ones, as defined in 2018 by the first author and Zhou, for which partial classification results are known. In this paper we propose a unified algebraic description of F(q)n-linear maximum rank distance codes, introducing the notion of exceptional linear maximum rank distance codes of a given index. Such a connection naturally extends the notion of exceptionality for a scattered polynomial in the rank metric framework and provides a generalization of Moore sets in the monomial mrd context. We move towards the classification of exceptional linear mrd codes, by showing that the ones of index zero are generalized Gabidulin codes and proving that in the positive index case the code contains an exceptional scattered polynomial of the same index.

关键词： Rank metric code mrd code scattered polynomial Moore set

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New Constructions of Subspace codes Using Subsets of mrd codes in Several Blocks

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IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第9期66卷 5317-5321页

作者： Chen, Hao He, Xianmang Weng, Jian Xu, Liqing Jinan Univ Coll Informat Sci & Technol Guangzhou 510632 Peoples R China Jinan Univ Coll Cyber Secur Guangzhou 510632 Peoples R China Dongguan Univ Technol Sch Network Secur Dongguan 523808 Peoples R China

A basic problem for the constant dimension subspace coding is to determine the maximal possible size A(q)(n, d, k) of a set of k-dimensional subspaces in F-q(n) such that the subspace distance satisfies d(U, V) = 2k - 2dim (U boolean AND V) >= d for any two different subspaces U and V in this set. We present two new constructions of constant dimension subspace codes using subsets of maximum rank-distance (mrd) codes in several blocks. This method is firstly applied to the linkage construction and secondly to arbitrary number of blocks of lifting mrd codes. In these two constructions, subsets of mrd codes with bounded ranks play an essential role. The Delsarte theorem about the rank distribution of mrd codes is an important ingredient to count codewords in our constructed constant dimension subspace codes. We give many new lower bounds for A(q)(n, d, k). More than 110 new constant dimension subspace codes better than previously best known codes are constructed

关键词： Constant dimension subspace code mrd code q-polynmial

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Constructions of rank metric codes under actions of the unitary groups

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LINEAR ALGEBRA AND ITS APPLICATIONS 2022年 645卷 293-306页

作者： Chen, Shangdi Xu, Qin Zhang, Junmei Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

Some rank metric codes are constructed based on the actions of cyclic unitary groups and Abelian noncyclic unitary groups. All the rank metric codes studied in this paper consist of proper rectangular matrices. Most o... 详细信息

关键词： Rank metric code orbit code mrd code Unitary group

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New Construction for Constant Dimension Subspace codes via a Composite Structure

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IEEE COMMUNICATIONS LETTERS 2021年第5期25卷 1422-1426页

作者： He, Xianmang Chen, Yindong Zhang, Zusheng Zhou, Kunxiao Dongguan Univ Technol Sch Cyberspace Secur Dongguan 526808 Peoples R China Shantou Univ Dept Comp Sci Shantou 515063 Peoples R China Shantou Univ Key Lab Intelligent Mfg Technol Minist Educ Shantou 515063 Peoples R China

One of the most fundamental topics in subspace coding is to explore the maximal possible value A(q)(n, d, k) of a set of k-dimensional subspaces in F-q(n) such that the subspace distance satisfies d(S)(U, V) = dim(U+V)-dim(U boolean AND V) >= d for any two different k-dimensional subspaces U and V in this set. In this letter, we propose a construction for constant dimension subspace codes by inserting a composite structure composing of an mrd code and its sub-codes. Its vast advantage over the previous constructions has been confirmed through extensive examples. At least 49 new constant dimension subspace codes which exceeds the currently best codes are constructed.

关键词： Measurement Couplings Upper bound Encoding Web pages Technology planning Manufacturing Subspace coding constant dimension codes projective space mrd code

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A New Family of mrd codes in Fq^2nx2n With Right and Middle Nuclei Fqn^q

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IEEE TRANSACTIONS ON INFORMATION THEORY 2019年第2期65卷 1054-1062页

作者： Trombetti, Rocco Zhou, Yue Univ Napoli Federico II Dipartimento Matemat & Applicaz R Caccioppoli I-80126 Naples Italy Natl Univ Def Technol Coll Liberal Arts & Sci Changsha 410073 Hunan Peoples R China

In this paper, we present a new family of maximum rank-distance (mrd) codes in F-q(2nx2n) of minimum distance 2 <= d <= 2n. In particular, when d = 2n, we can show that the corresponding semifield is exactly a Hughes-Kleinfeld semifield. The middle and right nuclei of these mrd codes are both equal to F-qn. We also prove that the mrd codes of minimum distance 2 < d < 2n in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.

关键词： Rank-metric code mrd code semifield Gabidulin code

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Almost affinely disjoint subspaces and covering Grassmannian codes

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FINITE FIELDS AND THEIR APPLICATIONS 2023年第1期92卷

作者： Yu, Shuhui Ji, Lijun Soochow Univ Dept Math Suzhou 215006 Peoples R China

A family of k-dimensional subspaces of Fqn with pairwise trivial intersection is called L-almost affinely disjoint (AAD) if each affine coset of a member of this family intersects with only at most L subspaces from the family. Liu et al. introduced the notion of AAD family, and investigated the lower and upper bounds of maximal such sets and conjectured that for any k, n and a large enough L = L(n, k), there exists an [n, k, L]q-AAD family with size qn-2k in [Finite Fields Appl. 75 (2021), 101879]. Etzion and Zhang introduced covering Grassmannian codes (CGCs) for generalized combination networks [IEEE Trans. Inf. Theory, 65 (2019), 4131-4142.]. An alpha-(n, k, delta)cq covering Grassmannian code C is a set of k-dimensional subspaces of Fqn, such that every set of alpha codewords of C spans a subspace of dimension at least k + delta. In this paper, we give a construction of AAD families and CGCs based on maximum rank metric codes and caps in projective geometries, and a recursive construction based on maximum rank metric codes. As a consequence, we prove that Liu et al.'s conjecture is still true for k >= 3 and n = 4k, and improve lower bounds on maximum sizes of AAD families and lower bounds on maximum sizes of 3-(n, k, 2k)cq covering Grassmannian codes.(c) 2023 Elsevier Inc. All rights reserved.

关键词： Almost affinely disjoint subspace Partial spread mrd code Cap Covering Grassmannian code Independent configuration

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mrd codes with maximum idealizers

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DISCRETE MATHEMATICS 2020年第9期343卷 111985-111985页

作者： Csajbok, Bence Marino, Giuseppe Polverino, Olga Zhou, Yue Eotvos Lorand Univ MTA ELTE Geometr & Algebra Combinator Res Grp Budapest Hungary Dept Geometry Pazmany P Stny 1-C H-1117 Budapest Hungary Univ Napoli Federico II Dipartimento Matemat & Applicaz Renato Caccioppol Via Cintia I-80126 Naples Italy Univ Campania Luigi Vanvitelli Dipartimento Matemat & Fis Viale Lincoln 5 I-81100 Caserta Italy Natl Univ Def Technol Coll Liberal Arts & Sci Changsha 410073 Peoples R China

Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (mrd for short) codes in F-q(nxn) the idealizers have been proved to be isomorphic to finite fields of size at most q(n). Up to now, the only known mrd codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify mrd codes in F-q(nxn) for n <= 9 with maximum left and right idealizers and connect them to Moore-type matrices. Apart from generalized Gabidulin codes, it turns out that there is a further family of rank-distance codes providing mrd ones with maximum idealizers for n = 7, q odd and for n = 8, q 1 (mod 3). These codes are not equivalent to any previously known mrd code. Moreover, we show that this family of rank-distance codes does not provide any further examples for n >= 9. (C) 2020 Elsevier B.V. All rights reserved.

关键词： mrd code Idealizer, Gabidulin code linearized polynomial Moore matrix Algebraic curve

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