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Several classes of linear codes with AMDS duals and their subfield codes

CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2024年第6期16卷 1429-1448页

作者： Qiao, Xingbin Du, Xiaoni Yuan, Wenping Northwest Normal Univ Coll Math & Stat Lanzhou 730070 Peoples R China Northwest Normal Univ Key Lab Cryptog & Data Analyt Lanzhou 730070 Peoples R China Gansu Prov Res Ctr Basic Disciplines Math & Stat Lanzhou 730070 Peoples R China

Subfield codes of linear codes over finite fields have recently attracted great attention due to their wide applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction of 3-dimensional linear codes C-f over finite field F-2(m) parameterized by any Boolean function f. Then we determine explicitly the weight distributions of C-f, the punctured code (C) over tilde (f), as well as the corresponding subfield codes over F-2 for several classes of Boolean functions ff. In particular, we determine the weight distributions of subfield codes derived from r-plateaued functions. Moreover, the parameters of their dual codes are investigated, which contain length-optimal and dimension-optimal AMDS codes with respect to the sphere packing bound. We emphasize that the new codes are projective and contain binary self-complementary codes. As applications, some of the projective codes we present can be employed to construct s-sum sets for any odd integer s >1.

关键词： Linear code Punctured code Subfield code projective code Weight distribution

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Some two-weight codes invariant under the 3-fold covers of the Mathieu groups M22 and Aut(M22)

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JOURNAL OF ALGEBRA AND ITS APPLICATIONS 2024年第7期23卷

作者： Rodrigues, B. G. Univ Pretoria Dept Math & Appl Math Private Bag X20 ZA-0028 Florida South Africa

Using an approach from finite group representation theory we construct quaternary non-projective codes with parameters [693,6,480]4,[1386,6,1008]4,[2016,6,1488]4, quaternary projective codes with parameters [231,6,160]4,[462,6,336]4 and [672,6,496]4 and binary projective codes with parameters [693,12,320]2,[1386,12,672]2,[2016,12,992]2 as examples of two-weight codes on which a finite almost quasisimple group of sporadic type acts transitively as permutation groups of automorphisms. In particular, we show that these codes are invariant under the 3-fold covers 3M22 and 3M22:2, respectively, of the Mathieu groups M22 and M22:2. Employing a known construction of strongly regular graphs from projective two-weight codes we obtain from the binary projective (respectively, quaternary projective) two-weight codes with parameters those given above, the strongly regular graphs with parameters (4096,693,152,110),(4096,1386,482,462), and (4096,2016,992,992), respectively. The latter graph can be viewed as a 2-(4096,2016,992)-symmetric design with the symmetric difference property whose residual and derived designs with respect to a block give rise to binary self-complementary codes meeting the Grey-Rankin bound with equality.

关键词： projective code two-weight code Mathieu group 3M(22) strongly regular graph

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Further projective binary linear codes derived from two-to-one functions and their duals

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DESIGNS codeS AND CRYPTOGRAPHY 2023年第3期91卷 719-746页

作者： Mesnager, Sihem Qian, Liqin Cao, Xiwang Univ Paris VIII Dept Math F-93526 St Denis France Univ Sorbonne Paris Cite CNRS LAGA UMR 7539 F-93430 Villetaneuse France Polytech Inst Paris Telecom Paris F-91120 Palaiseau France Nanjing Univ Aeronaut & Astronaut Dept Math Nanjing 211106 Jiangsu Peoples R China MIIT Key Lab Math Modelling & High Performance Comp Ai Nanjing 211106 Peoples R China

Binary linear codes with few weights have wide applications in communication, secret sharing schemes, authentication codes, association schemes, strongly regular graphs, etc. projective binary linear codes are among the most important subclasses of binary linear codes for practical applications. In this paper, motivated by the two excellent recent papers (Li et al. in IEEE Trans Inf Theory 67(7):4263-4275, 2021) and (Wang et al. in IEEE Trans Inf Theory 67(8):5133-5148, 2021), several new families of few-weight projective binary linear codes are constructed from the defining sets, and then their Hamming weight distributions are determined by employing the Walsh transform of the corresponding two-to-one functions over finite fields with the even characteristic. Our constructions can produce binary linear codes with new parameters. Some of the constructed binary linear codes are optimal or almost optimal according to the online Database of Grassl, and the duals of some of them are distance-optimal with respect to the sphere packing bound. This paper also shows once again that the two-to-one functions initially studied in (Mesnager and Qu in IEEE Trans Inf Theory 65(12):7884-7895, 2019) are also promising objects in coding theory. Although our derived codes use objects considered in the very recent literature, the analysis of our designed codes involves functions having different algebraic structures (and, therefore, other Walsh transform distribution) and requires solving new systems of equations over finite fields, which is an essential step in determining the weight distribution of our constructed codes. As applications, some of the codes presented in this paper can be used to construct association schemes and secret sharing schemes with interesting access structures.

关键词： projective code Hamming weight distribution Two-to-one function Association scheme Secret sharing scheme

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projective binary linear codes from special Boolean functions

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2021年第4期32卷 521-552页

作者： Heng, Ziling Wang, Weiqiong Wang, Yan Changan Univ Sch Sci Xian 710064 Shaanxi Peoples R China Xian Univ Architecture & Technol Sch Sci Xian 710055 Shaanxi Peoples R China

Linear codes with a few weights have nice applications in communication, secret sharing schemes, authentication codes, association schemes, block designs and so on. projective binary linear codes are one of the most important subclasses of linear codes for practical applications. The objective of this paper is to construct projective binary linear codes with some special Boolean functions. Four families of binary linear codes with three or four weights are derived and the parameters of their duals are also determined. It turns out that the duals of these codes are optimal or almost optimal with respect to the sphere-packing bound. As applications, the codes presented in this paper can be used to construct association schemes and secret sharing schemes with interesting access structures.

关键词： Boolean function Linear code projective code Weight distribution

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Constructions of projective linear codes by the intersection and difference of sets

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FINITE FIELDS AND THEIR APPLICATIONS 2022年第0期83卷

作者： Heng, Ziling Chen, Fuling Xie, Chunlei Li, Dexiang Changan Univ Sch Sci Xian 710064 Peoples R China Changan Univ Sch Informat Engn Xian 710064 Peoples R China

projective linear codes are a special class of linear codes whose dual codes have minimum distance at least 3. projective linear codes with only a few weights are useful in authentication codes, secret sharing schemes, data storage systems and so on. In this paper, two constructions of q-ary linear codes are presented with defining sets given by the intersection and difference of two sets. These constructions produce several families of new projective two-weight or three-weight linear codes. As applications, our projective codes can be used to construct secret sharing schemes with interesting access structures, strongly regular graphs and association schemes with three classes.(c) 2022 Elsevier Inc. All rights reserved.

关键词： Linear code projective code Secret sharing scheme Strongly regular graph

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The Lengths of projective Triply-Even Binary codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第5期66卷 2713-2716页

作者： Honold, Thomas Kiermaier, Michael Kurz, Sascha Wassermann, Alfred Zhejiang Univ ZJU UIUC Inst Int Campus Haining 314400 Peoples R China Univ Bayreuth Dept Math Phys & Comp Sci D-95447 Bayreuth Germany

It is shown that there does not exist a projective triply-even binary code of length 59. This settles the last open length for projective triply-even binary codes, which therefore exist precisely for the lengths 15, 1... 详细信息

关键词： Linear codes Generators Binary codes Upper bound Hamming weight projective geometry Divisible code projective code linear code partial spread

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The graphs of projective codes

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FINITE FIELDS AND THEIR APPLICATIONS 2018年 54卷 15-29页

作者： Kwiatkowski, Mariusz Pankov, Mark Pasini, Antonio Univ Warmia & Mazury Fac Math & Comp Sci Sloneczna 54 Olsztyn Poland Univ Siena Dept Informat Engn & Math Via Roma 56 Siena Italy

Consider the Grassmann graph formed by k-dimensional subspaces of an n-dimensional vector space over the field of q elements (1 = ((n)(2)), we show that the graph II(n, k)(q) is connected, its diameter is equal to the... 详细信息

Consider the Grassmann graph formed by k-dimensional subspaces of an n-dimensional vector space over the field of q elements (1 < k < n-1) and denote by II(n, k), the restriction of this graph to the set of projective [n, k](q) codes. In the case when q >= ((n)(2)), we show that the graph II(n, k)(q) is connected, its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph. Also, we give some observations concerning the graphs of simplex codes. For example, binary simplex codes of dimension 3 are precisely maximal singular subspaces of a non-degenerate quadratic form. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Linear code projective code Simplex code Grassmann graph

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Implementing line-Hermitian Grassmann codes

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LINEAR ALGEBRA AND ITS APPLICATIONS 2019年 580卷 96-120页

作者： Cardinali, Ilaria Giuzzi, Luca Univ Siena Dept Informat Engn & Math Via Roma 56 I-53100 Siena Italy Univ Brescia Sect Math DICATAM Via Branze 43 I-125123 Brescia Italy

In [6] we introduced line Hermitian Grassmann codes and determined their parameters. The aim of this paper is to present (in the spirit of [4]) an algorithm for the point enumerator of a line Hermitian Grassmannian which can be usefully applied to get efficient encoders, decoders and error correction algorithms for the aforementioned codes. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Hermitian variety Polar Grassmannian projective code Point enumerator

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Line Hermitian Grassmann codes and their parameters

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FINITE FIELDS AND THEIR APPLICATIONS 2018年 51卷 407-432页

作者： Cardinali, Ilaria Giuzzi, Luca Univ Siena Dept Informat Engn & Math Via Roma 56 I-53100 Siena Italy Univ Brescia Sect Math DICATAM Via Branze 43 I-25123 Brescia Italy

In this paper we introduce and study line Hermitian Grassmann codes as those subcodes of the Grassmann codes associated to the 2-Grassmannian of a Hermitian polar space defined over a finite field. In particular, we d... 详细信息

关键词： Hermitian variety Polar Grassmannian projective code

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A Characterization of Two-Weight projective Cyclic codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2015年第1期61卷 66-71页

作者： Feng, Tao Zhejiang Univ Dept Math Hangzhou 310027 Peoples R China Beijing Ctr Math & Informat Interdisciplinary Sci Beijing 100048 Peoples R China

We give necessary conditions for a two-weight projective cyclic code to be the direct sum of two one-weight irreducible cyclic subcodes of the same dimension, following the work of Wolfmann and Vega. This confirms Veg... 详细信息

关键词： Cyclic code two-weight code projective code Gauss sum

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