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Generalized bilateral multilevel construction for constant dimension codes

DESIGNS codeS AND CRYPTOGRAPHY 2025年第1期93卷 197-225页

作者： Hong, Xiaoqin Cao, Xiwang Luo, Gaojun Nanjing Univ Aeronaut & Astronaut NUAA Sch Math Nanjing 211106 Peoples R China Key Lab Math Modelling & High Performance Comp Air Nanjing 211106 Peoples R China

constant dimension codes (CDCs) have drawn extensive attention due to their applications in random network coding. This paper introduces a new class of codes, namely generalized bilateral Ferrers diagram rank-metric codes, to generalize the bilateral multilevel construction in Etzion and Vardy (Adv Math Commun 16:1165-1183, 2022). Combining our generalized bilateral multilevel construction and the double multilevel construction in Liu and Ji (IEEE Trans Inf Theory 69:157-168, 2023), we present an effective technique to construct CDCs. By means of bilateral identifying vectors, this approach helps us to select fewer identifying and inverse identifying vectors to construct CDCs with larger size. The new constructed CDCs have the largest size regarding known codes for many sets of parameters. Our method gives rise to at least 138 new lower bounds for CDCs.

关键词： constant dimension code Rank-metric code Identifying vector Ferrers diagram Multilevel construction

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New constant dimension subspace codes via generalized coset construction

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2025年 1-13页

作者： Singh, Kanchan Singh, Sheo Kumar Mahatma Gandhi Cent Univ Dept Math Motihari 845401 Bihar India

The fundamental problem of subspace coding is to explore the maximum possible cardinality A(q)(n, d, k) of a set of k-dimensional subspaces of an n-dimensional vector space over F(q )such that the subspace distance satisfies d(S)(W-1, W-2) >= d for any two distinct subspaces W-1,W(2 )in this set. In this paper, we construct a new class of constant dimension codes (CDCs) by generalizing the coset construction and combining it with CDCs derived from parallel linkage construction and coset construction with an aim to improve the new lower bounds of A(q)(n, d, k). We found a remarkable improvement in some of the lower bounds of A(q)(n, d, k).

关键词： constant dimension code Rank-metric code Coset construction Parallel linkage construction

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Generalized block inserting for constructing new constant dimension codes

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2023年第1期15卷 1-15页

作者： Hong, Xiaoqin Cao, Xiwang Nanjing Univ Aeronaut & Astronaut Coll Math Nanjing 211106 Peoples R China MIIT Key Lab Math Modelling & High Performance Comp Ai Nanjing 211106 Peoples R China

constant dimension codes (CDCs) have drawn extensive attention due to their applications in random network coding. A fundamental problem for CDCs is to explore the maximum possible cardinality A(q)(n,d,k) of a set of k-dimensional subspaces in F-q(n) such that the sub-space distance statisfies dis(U,V) = 2k - 2 dim(U boolean AND V) >= d for all pairs of distinct sub-spaces U and V in this set. In this paper, by means of an appropriate combination of the matrix blocks from rank metric codes and small CDCs, we present three constructions of CDCs based on the generalized block inserting construction by Niu et al. in 2021. According to our constructions, we obtain 28 new lower bounds for CDCs which are better than the previously known lower bounds.

关键词： Subspace coding constant dimension code Inserting construction Rank metric code

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Double Multilevel Constructions for constant dimension codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2023年第1期69卷 157-168页

作者： Liu, Shuangqing Ji, Lijun Suzhou Univ Sci & Technol Dept Math Suzhou 215009 Peoples R China Soochow Univ Dept Math Suzhou 215006 Peoples R China

constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called Ferrers diagram rank-metric codes with given ranks (GFRMCs), to generalize the parallel construction and the parallel multilevel construction in [IEEE Trans. Inf. Theory, 66 (2020), 6884-6897]. The lower bounds for GFRMCs are derived from Ferrers diagram rank-metric codes (FDRMCs). Via GFRMCs, the inverse multilevel construction for CDCs is showed. Furthermore, the double multilevel construction, as an effective construction for CDCs, is presented by combining the inverse multilevel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given.

关键词： constant dimension code multilevel construction parallel construction rank-metric code Ferrers diagram

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Parallel Multilevel Constructions for constant dimension codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第11期66卷 6884-6897页

作者： Liu, Shuangqing Chang, Yanxun Feng, Tao Beijing Jiaotong Univ Dept Math Beijing 100044 Peoples R China

constant dimension codes (CDCs), as special sub-space codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called rank metric codes with given ranks (GRMCs), to generalize the parallel construction in [Xu and Chen, IEEE Trans. Inf. Theory, 64 (2018), 6315-6319] and the classic multilevel construction. A Singleton-like upper bound and a lower bound for GRMCs derived from Gabidulin codes are given. Via GRMCs, two effective constructions for CDCs are presented by combining the parallel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given. The ratio between the new lower bound and the known upper bound for (4 delta, 2 delta, 2 delta)q(-) CDCs is calculated. It is greater than 0.99926 for any prime power q and any delta >= 3.

关键词： constant dimension code rank-metric code multilevel construction parallel construction

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A Note on Optimal constant dimension codes

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INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE 2015年第1期26卷 143-152页

作者： Liao, Qunying Zhu, Juan Sichuan Normal Univ Inst Math & Software Sci Chengdu 610066 Sichuan Peoples R China

In this paper, we study bounds for optimal constant dimension codes furl By the construction for constant dimension codes in [4], we improve some bounds on q-ary constant dimension codes in some cases. By combinatorial method, we show that there exists no optimal constant dimension code A(q)[n, 2 delta, k] meeting both Wang-Xing-Safavi-Naini-Bound and the maximal distance separate bound simultaneously.

关键词： constant dimension code dimension distance lower bound upper bound

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Equivalence for flag codes

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LINEAR ALGEBRA AND ITS APPLICATIONS 2024年 690卷 1-26页

作者： Navarro-Perez, Miguel angel Soler-Escriva, Xaro Univ Carlos III Madrid Dept Matemat Ave Univ 30 Leganes 28911 Madrid Spain Univ Alacant Dept Matemat Ap Correus 99 Alacant 03080 Spain

Given a finite field F q and a positive integer n, a flag is a sequence of nested F q -subspaces of a vector space F q n and a flag code is a nonempty collection of flags. The projected codes of a flag code are the constant dimension codes containing all the subspaces of prescribed dimensions that form the flags in the flag code. In this paper we address the notion of equivalence for flag codes and explore in which situations such an equivalence can be reduced to the equivalence of the corresponding projected codes. In addition, this study leads to new results concerning the automorphism group of certain families of flag codes, some of them also introduced in this paper. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY -NC license (http:// creativecommons .org /licenses /by -nc /4 .0/).

关键词： Flag codes constant dimension code (semi)linear equivalence Automorphism group

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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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Optimal Ferrers diagram rank-metric codes from MRD codes

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DESIGNS codeS AND CRYPTOGRAPHY 2023年第12期91卷 3977-3993页

作者： Liu, Shuangqing Suzhou Univ Sci & Technol Dept Math Suzhou 215009 Peoples R China

Subspace codes, motivated by their extensive application in random network coding, have become one of central topics in algebraic coding theory during the last 10 years. Due to the significant application in subspace codes, Ferrers diagram rank-metric (FDRM) codes also have drawn a lot of attention. In this paper, we introduce two new constructions based on subcodes ofMRD codes. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. Apply the first construction to solve the optimality of [F, 4]q-FDRM codes, where F = [2, 2, 4, 4,..., 2l, 2l], which was raised in Etzion et al. (IEEE Trans Inf Theory 62:1616-1630, 2016). By the restricted Gabidulin codes and improving the way to select the subcodes, the second construction is presented, which unifies and generalizes all known constructions based on subcodes of Gabidulin codes. By the second construction, we can give new families of optimal FDRM codes, whose numbers of codewords are unequal to qv0. This paper also shows new families of FDRM codes whose optimality cannot be obtained by the constructions based on subcodes of Fqm -linear MRD codes.

关键词： Ferrers diagram Rank-metric code Gabidulin code constant dimension code

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Several classes of optimal Ferrers diagram rank-metric codes

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LINEAR ALGEBRA AND ITS APPLICATIONS 2019年 581卷 128-144页

作者： Liu, Shuangqing Chang, Yanxun Feng, Tao Beijing Jiaotong Univ Dept Math Beijing 100044 Peoples R China

Four constructions for Ferrers diagram rank-metric (FDRM) codes are presented. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. By introducing restricted Gabidulin codes, the second construction is presented, which unifies many known constructions for FDRM codes. The third and fourth constructions are based on two different ways to represent elements of a finite field Fe, (vector representation and matrix representation). Each of these constructions produces optimal codes with different diagrams and parameters. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Ferrers diagram Rank-metric code Gabidulin code constant dimension code

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