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subspace code based on flats in affine space over finite fields

DISCRETE MATHEMATICS ALGORITHMS AND APPLICATIONS 2018年第6期10卷

作者： Niu, Min-Yao Wang, Gang Gao, You Fu, Fang-Wei Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China Nankai Univ Chern Inst Math Tianjin 300071 Peoples R China Nankai Univ LPMC Tianjin 300071 Peoples R China

subspace code is very useful in the error correction for random network coding. In this paper, subspace code based on flats in affine space over finite fields is constructed. Sphere-packing bound, Wang-Xing-Safavi-Naini bound, anticode bound, Ahlswede-Aydinian bound and Gilbert-Varshamov bound for the size of code based on flats in affine space over finite fields are provided. Finally, we introduce another method of obtaining the anticode bound for the size of code based on flats in affine space from Erd&-Ko-Rado theorem.

关键词： subspace code affine space bound Erdos-Ko-Rado theorem finite fields

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Vector multispaces and multispace codes

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APPLIED MATHEMATICS AND COMPUTATION 2025年 486卷

作者： Kovacevic, Mladen Univ Novi Sad Fac Tech Sci Novi Sad Serbia

Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than 1 are derived. An application in coding theory is illustrated by showing that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes (defined in the set of all subspaces of F-q(n)) and extend them to an infinitely larger set of parameters. In particular, in contrast to subspace codes, multispace codes of arbitrarily large cardinality and minimum distance exist for any fixed n and q.

关键词： Multiset Multispace Projective space Grassmannian Linearized polynomial Modular lattice subspace code Random linear network coding

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New constructions of abelian non-cyclic orbit codes based on parabolic subgroups and tensor products

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FINITE FIELDS AND THEIR APPLICATIONS 2025年 103卷

作者： Askary, Soleyman Biranvand, Nader Shirjian, Farrokh Imam Ali Univ Fac Sci Tehran Iran

Orbit codes, as special constant dimension subspace codes, have attracted much attention due to their applications for error correction in random network coding. They arise as orbits of a subspace of Fqn under the action of some subgroup of the finite general linear group GLn(q). The main contribution of this paper is to propose new methods for constructing large non-cyclic orbit codes. First, using the subgroup structure of maximal subgroups of GLn(q), we propose a new construction of an abelian non-cyclic orbit codes of size qk with k <= n/2. The proposed code is shown to be a partial spread which in many cases is close to the known maximum-size codes. Next, considering a larger framework, we introduce the notion of tensor product operation for subspace codes and explicitly determine the parameters of such product codes. The parameters of the constructions presented in this paper improve the constructions already obtained in [6] and [7]. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Random linear network coding Algebraic coding theory Abelian orbit code subspace code Linear groups

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New constructions of constant dimension subspace codes with large sizes

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DESIGNS codeS AND CRYPTOGRAPHY 2024年第5期92卷 1423-1437页

作者： Li, Yun Liu, Hongwei Mesnager, Sihem Cent China Normal Univ Sch Math & Stat Wuhan 430079 Peoples R China Univ Paris VIII Dept Math LAGA F-91120 Paris France Univ Paris XIII CNRS UMR 7539 F-91120 Palaiseau France Telecom ParisTech Palaiseau France

subspace codes have important applications in random network coding. It is a classical problem to construct subspace codes where both their size and their minimum distance are as large as possible. In particular, cyclic constant dimension subspace codes have additional properties which can be used to make encoding and decoding more efficient. In this paper, we construct large cyclic constant dimension subspace codes with minimum distances 2k-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2k-2$$\end{document} and 2k. These codes are contained in Gq(n,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}_q(n, k)$$\end{document}, where Gq(n,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}_q(n, k)$$\end{document} denotes the set of all k-dimensional subspaces of the finite filed Fqn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q<^>n}$$\end{document} of qn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>n$$\end{document} elements (q a prime power). Consequently, some results in [7, 15], and [23] are extended.

关键词： Sidon space subspace code Cyclic constant dimension code

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LCD subspace codes

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DESIGNS codeS AND CRYPTOGRAPHY 2023年第10期91卷 3215-3226页

作者： Crnkovic, Dean Svob, Andrea Univ Rijeka Fac Math Radmile Matejcic 2 Rijeka 51000 Croatia

A subspace code is a nonempty set of subspaces of a vector space F-q(n). Linear codes with complementary duals, or LCD codes, are linear codes whose intersection with their duals is trivial. In this paper, we introduce a notion of LCD subspace codes. We show that the minimum distance decoding problem for an LCD subspace code reduces to a problem that is simpler than for a general subspace code. Further, we show that under some conditions equitable partitions of association schemes yield such LCD subspace codes and as an illustration of the method give some examples from distance-regular graphs. We also give constructions from mutually unbiased weighing matrices, that include constructions from mutually unbiased Hadamard matrices.

关键词： Association scheme Equitable partition Mutually unbiased Hadamard matrices LCD code subspace code

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Further constructions of cyclic subspace codes

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2021年第2期13卷 245-262页

作者： Zhang, He Cao, Xiwang Nanjing Univ Aeronaut & Astronaut Dept Math Nanjing 211100 Peoples R China Chinese Acad Sci Inst Informat Engn State Key Lab Informat Secur Beijing 100093 Peoples R China

subspace codes, especially cyclic subspace codes, have attracted a wide attention in the past few decades due to their applications in error correction for random network coding. In 2016, Ben-Sasson et al. gave a systematic approach to constructing cyclic subspace codes by employing subspace polynomials. Inspired by Ben-Sasson's idea, Chen et al. also provided some constructions of cyclic subspace codes in 2017. In this paper, two constructions of cyclic subspace codes are given to further improve the results of Chen and Roth et al. respectively. Consequently, we obtain more cyclic subspace codes with larger size of codewords without reducing the minimum distance.

关键词： Constant dimension code subspace polynomial Random network coding Linearized polynomial subspace code Sidon space

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On subspace codes

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DESIGNS codeS AND CRYPTOGRAPHY 2016年第2期78卷 527-531页

作者： Cossidente, Antonio Pavese, Francesco Univ Basilicata Dipartimento Matemat & Informat Contrada Macchia Romana I-85100 Potenza Italy

It is shown that any projective bundle of PG(2, q) gives rise to a q-ary (6, q(6) + 2q(2) + 2q + 1, 4;3) subspace code.

关键词： Projective bundle subspace code Klein quadric Singer cyclic group

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subspace codes from Ferrers diagrams

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JOURNAL OF ALGEBRA AND ITS APPLICATIONS 2017年第7期16卷 1750131-1750131页

作者： Gorla, Elisa Ravagnani, Alberto Univ Neuchatel Inst Math Rue Emile Argand 11 CH-2000 Neuchatel Switzerland

In this paper, we survey the main known constructions of Ferrers diagram rank-metric codes, and establish new results on a related conjecture by Etzion and Silberstein. We also give a sharp lower bound on the dimension of linear rank-metric anticodes with a given profile. Combining our results with the multilevel construction, we produce examples of subspace codes with the largest known cardinality for the given parameters. We also apply results from algebraic geometry to the study of the analogous problem over an algebraically closed field, proving that the bound by Etzion and Silberstein can be improved in this case, and providing a sharp bound for full-rank matrices.

关键词： Matrix linear space rank subspace code

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Theory and Applications of Network Error Correction Coding

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PROCEEDINGS OF THE IEEE 2011年第3期99卷 406-420页

作者： Zhang, Zhen Univ So Calif Inst Commun Sci Dept Elect Engn Syst Los Angeles CA 90089 USA

Network error correction coding (NEC) has attracted a lot of attention in recent years because of its potential usefulness in network communications. Several kinds of errors may occur in communication networks using network coding. This includes random errors, erasures, and errors caused by attacks from malicious nodes. The main goal of the theory of NEC is to deal with these errors efficiently. Two kinds of network models have been considered in network coding theory: coherent and noncoherent networks. A network is called coherent if network characteristics are known to senders and receivers, and called noncoherent if they are unknown. Although both scenarios are theoretically interesting, the noncoherent network model fits better to the requirements in most applications. So far, there are two lines of research in the theory of NEC. One approach follows the classical method by representing messages by sequences and the other approach uses the theory of rank metric codes for network error correction by representing messages by subspaces. Following both approaches, basic theories have been developed. This includes the formulation of channel models, the characterization of error correction/detection capabilities for various kinds of errors, the derivation of bounds for NECs, the study of existence and constructions of optimal codes, the development of encoding and decoding techniques, and the design of code suitable for real applications. In this paper, we summarize some important contributions in this research direction.

关键词： Attacks from malicious nodes decoding algorithms erasures maximum distance separable code network coding network error correction coding (NEC) product code rank metric code singleton bound subspace code

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Constructions of optimal Ferrers diagram rank metric codes

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DESIGNS codeS AND CRYPTOGRAPHY 2019年第1期87卷 107-121页

作者： Zhang, Tao Ge, Gennian Guangzhou Univ Sch Math & Informat Sci Guangzhou 510006 Guangdong Peoples R China

subspace codes and constant dimension codes have become a widely investigated research topic due to their significance to error control in random linear network coding. Rank metric codes in Ferrers diagrams can be used to construct good subspace codes and constant dimension codes. In this paper, three constructions of Ferrers diagram rank metric codes are presented. The first two constructions are based on subcodes of maximum rank distance codes, and the last one generates new codes from known Ferrers diagram rank metric codes. Each of these constructions produces optimal codes with different diagrams and parameters for which no optimal construction was known before.

关键词： Ferrers diagram Rank metric code Gabidulin code subspace code Constant dimension code 15A03 15A99 15B99

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