检索结果-内蒙古大学图书馆

On some polynomials related to weight enumerators of linear codes

SIAM JOURNAL ON DISCRETE MATHEMATICS 2002年第2期15卷 155-164页

作者： Barg, A Bell Labs Lucent Technol Murray Hill NJ 07974 USA

A linear code can be thought of as a vector matroid represented by the columns of the code's generator matrix;a well-known result in this context is Greene's theorem on a connection of the weight polynomial of the code and the Tutte polynomial of the matroid. W examine this connection from the coding-theoretic viewpoint, building upon the rank polynomial of the code. This enables us to obtain bounds on all-terminal reliability of linear matroids and new proofs of two known results: Greene's theorem and a connection between the weight polynomial and the partition polynomial of the Potts model.

关键词： all-terminal reliability Greene's theorem linear code linear matroid Potts model rank polynomial Tutte polynomial weight numerator

来源：评论

学校读者我要写书评

暂无评论

On the minimum length of some linear codes

引用

DESIGNS codeS AND CRYPTOGRAPHY 2007年第2-3期43卷 123-135页

作者： Cheon, E. J. Maruta, T. Gyeongsang Natl Univ Dept Math Jinju 660701 South Korea Osaka Prefecture Univ Dept Math & Informat Sci Osaka 5998531 Japan

We determine the minimum length n(q) (k, d) for some linear codes with k >= 5 and q >= 3. We prove that n(q) (k, d) = g(q) (k, d) + 1 for q(k-1) - 2q k-1/2 - q + 1 <= d <= q(k-1) - 2q k-1/2 when k is odd, for q(k-1) q k/2 k/2 -1 -q+1 <= d <= q(k-1)-q(k/2) - q k2/1 when k is even, and for 2q(k-1) - 2q(k-2) - q(2) - q + 1 <= d <= 2q(k-1) - 2q(k-2) - q(2).

关键词： Griesmer bound linear code 0-cycle minimum length minihypers projective space

来源：评论

学校读者我要写书评

暂无评论

The projective general linear group PGL(2, 2^m) and linear codes of length 2^m+1

引用

DESIGNS codeS AND CRYPTOGRAPHY 2021年第7期89卷 1713-1734页

作者： Ding, Cunsheng Tang, Chunming Tonchev, Vladimir D. Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Kowloon Clear Water Bay Hong Kong Peoples R China China West Normal Univ Sch Math & Informat Nanchong 637002 Sichuan Peoples R China Michigan Technol Univ Dept Math Sci Houghton MI 49931 USA

Let q = 2(m). The projective general linear group PGL(2, q) acts as a 3-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over GF(2(h)) that are invariant under PGL(2, q) are trivial codes: the repetition code, the whole space GF(2(h))2(m)+1, and their dual codes. As an application of this result, the 2-ranks of the (0,1)-incidence matrices of all 3 - (q + 1, k, lambda) designs that are invariant under PGL(2, q) are determined. The second objective is to present two infinite families of cyclic codes over GF(2(m)) such that the set of the supports of all codewords of any fixed nonzero weight is invariant under PGL(2, q), therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters [q + 1, q - 3, 4] q, where q = 2(m), and m >= 4 is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3-(q + 1, 4, 2) design. A code from the second family has parameters [q + 1, 4, q - 4] q, q = 2(m), m >= 4 even, and the minimum weight codewords support a 3-(q + 1, q - 4, (q - 4)(q - 5)(q - 6)/60) design, whose complementary 3-(q + 1, 5, 1) design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over GF(q) that can support a 3-(q + 1, q - 4, (q - 4)(q - 5)(q - 6)/60) design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound.

关键词： Cyclic code linear code t-design Projective general linear group Automorphism group

来源：评论

学校读者我要写书评

暂无评论

Some Punctured codes of Several Families of Binary linear codes

引用

IEEE TRANSACTIONS ON INFORMATION THEORY 2021年第8期67卷 5133-5148页

作者： Wang, Xiaoqiang Zheng, Dabin Ding, Cunsheng Hubei Univ Fac Math & Stat Hubei Key Lab Appl Math Wuhan 430062 Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China

Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(f) = {Tr(af (x) + bx) x is an element of F-qm* :a, b is an element of F-q(m)}, where q is a prime power, F-q(m) = F-q(m) \ linear code, Tr is the trace function from F-q(m) to F-q, and f (x) is a function from F-q(m) to F-q(m) with f (0) = 0. Almost bent functions, quadratic functions and some monomials on F-2(m) were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper.

关键词： Boolean function linear code punctured code shortened code weight distribution

来源：评论

学校读者我要写书评

暂无评论

A class of optimal linear codes of length one above the Griesmer bound

引用

DESIGNS codeS AND CRYPTOGRAPHY 2009年第1期51卷 9-20页

作者： Cheon, E. J. Gyeongsang Natl Univ Dept Math Jinju 660701 South Korea Gyeongsang Natl Univ RINS Jinju 660701 South Korea

In this paper, we determine the smallest lengths of linear codes with some minimum distances. We construct a [g(q)(k, d) + 1, k,d](q) code for sq(k-1) - sq(k-2) - q(s) - q(2)+ 1 = s + 1. Then we get n(q) (k,d) = g(q)(... 详细信息

In this paper, we determine the smallest lengths of linear codes with some minimum distances. We construct a [g(q)(k, d) + 1, k,d](q) code for sq(k-1) - sq(k-2) - q(s) - q(2)+ 1 <= d <= sq(k-1) - sq(k-2) - q(s) with 3 <= s <= k-2 and q >= s + 1. Then we get n(q) (k,d) = g(q)(k,d) + 1 for (k -2)q(k-1) - (k-1)q(k-2) - q(2) + 1 <= d <= (k -2)q(k-1) - (k-1)q(k-2), k >= 6, q >= 2k-3;and sq(k-1) - sq(k-2) - q(s) - q + 1 <= d <= sq(k-1) - sq(k-2) - q(s), s >= 2, k >= 2s + 1 and q >= 2s -1.

关键词： Griesmer bound linear code 0-cycle Minimum length Minihyper Projective space

来源：评论

学校读者我要写书评

暂无评论

The minimum locality of linear codes

引用

DESIGNS codeS AND CRYPTOGRAPHY 2023年第1期91卷 83-114页

作者： Tan, Pan Fan, Cuiling Ding, Cunsheng Tang, Chunming Zhou, Zhengchun Yunnan Normal Univ Sch Informat Sci & Technol Kunming 650500 Yunnan Peoples R China Southwest Jiaotong Univ Sch Math Chengdu 611756 Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Kowloon Clear Water Bay Hong Kong Peoples R China China West Normal Univ Sch Math & Informat Nanchong 637002 Sichuan Peoples R China

Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a lack of general theory on the minimum locality of linear codes. In addition, the minimum locality of many known families of linear codes has not been studied in the literature. Motivated by these two facts, this paper develops some general theory about the minimum locality of linear codes, and investigates the minimum locality of a number of families of linear codes, such as q-ary Hamming codes, q-ary Simplex codes, generalized Reed-Muller codes, ovoid codes, maximum arc codes, the extended hyperoval codes, and near MDS codes. Many classes of both distance-optimal and dimension-optimal LRCs are presented in this paper. To this end, the concepts of linear locality and minimum linear locality are specified. The minimum linear locality of many families of linear codes are settled with the general theory developed in this paper.

关键词： Cyclic code linear code Locally recoverable code Near MDS code Punctured code Shortened code

来源：评论

学校读者我要写书评

暂无评论

Several Families of Binary Minimal linear codes From Two-to-One Functions

引用

IEEE TRANSACTIONS ON INFORMATION THEORY 2023年第5期69卷 3285-3301页

作者： Mesnager, Sihem Qian, Liqin Cao, Xiwang Yuan, Mu Univ Paris 08 Dept Math F-93526 St Denis France Univ Sorbonne Paris Cite LAGA UMR 7539 CNRS F-93430 Villetaneuse France Polytech Inst Paris Telecom Paris F-91120 Palaiseau France Anhui Univ Sch Math Sci Hefei 230601 Peoples R China Nanjing Univ Aeronaut & Astronaut Dept Math Nanjing 211106 Jiangsu Peoples R China

Minimal linear codes have important applications in secure communications, including in the framework of secret sharing schemes and secure multi-party computation. A lot of research have been carried out to derive codes with few weights (but more importantly, being minimal) using algebraic or geometric approaches. One of the main power and fructify algebraic methods is based on the design of those codes by employing functions over finite fields. Li et al. (2021) have recently identified some binary linear codes with few weights from two classes of two- to-one functions. In this paper, our ultimate objective is to expand the class of codes derived from the paper of Li et al. by proposing larger classes of binary linear codes with few weights via generic constructions involving other known families of two-to-one functions over the finite field F2(n) of order 2(n). We succeed in constructing such codes, and we also completely determine their weight distributions. The linear codes presented in this paper differ in parameters from those known in the literature. Besides, some of them are optimal concerning the well-known Griesmer bound. Notably, we prove that our codes are either optimal or almost optimal with respect to the online Database of Grassl. We next observe that the derived binary linear codes also have the minimality property for most cases. We then describe the access structures of the secret-sharing schemes based on their dual codes. Finally, we solve two problems left open in the paper by Li et al. (more specifically, a complete solution to Problem 2 and a partial solution to Problem 1).

关键词： linear code minimal code vectorial Boolean function two-to-one function secret sharing scheme hamming weight distribution

来源：评论

学校读者我要写书评

暂无评论

A class of three-weight and five-weight linear codes

引用

DISCRETE APPLIED MATHEMATICS 2018年 241卷 25-38页

作者： Li, Fei Wang, Qiuyan Lin, Dongdai Anhui Univ Finance & Econ Sch Stat & Appl Math Bengbu City 233030 Anhui Peoples R China Tianjin Polytech Univ Sch Comp Sci & Software Engn Tianjin 300387 Peoples R China Chinese Acad Sci Inst Informat Engn State Key Lab Informat Secur Beijing 100195 Peoples R China

Recently, linear codes with few weights have been widely studied, since they have applications in data storage systems, communication systems and consumer electronics. In this paper, we present a class of three-weight and five-weight linear codes over F-p, where p is an odd prime and F-p denotes a finite field with p elements. The weight distributions of the linear codes constructed in this paper are also settled. Moreover, the linear codes illustrated in the paper may have applications in secret sharing schemes. (C) 2016 Published by Elsevier B.V.

关键词： linear code Weight distribution Gaussian sums Weight enumerator Secret sharing

来源：评论

学校读者我要写书评

暂无评论

THE NONEXISTENCE OF CERTAIN BINARY linear codeS

引用

IEEE TRANSACTIONS ON INFORMATION THEORY 1990年第4期36卷 917-922页

作者： HILL, R TRAYNOR, KL Department of Mathematics and Computer Science University of Sanford Salford UK Belle Vue Rise Devon UK

Some of the upper bounds given by T. Verhoeff (1987) are improved by proving the nonexistence of codes with certain parameters. Necessary preliminary results are stated. The results give rise to upper bounds on d(n, k) for many further values of (n, k) by using simple standard techniques. The further bounds would automatically follow from theorems by using Verhoeff's computerized updating program, but the authors state them here because several of them are required in the proofs of subsequent theorems.< >

关键词： Upper bound Equations linear code Hamming weight Vectors Mathematics Computer science Parity check codes

来源：评论

学校读者我要写书评

暂无评论

Some results on linear codes over the ring Z4

引用

JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2017年第1-2期54卷 307-324页

作者： Li, Ping Guo, Xuemei Zhu, Shixin Kai, Xiaoshan Hefei Univ Technol Sch Math Hefei 230009 Anhui Peoples R China Southeast Univ Natl Mobile Commun Res Lab Nanjing 210096 Jiangsu Peoples R China

In this paper, we mainly study the theory of linear codes over the ring R = Z(4) + uZ(4) + nu Z(4) + u nu Z(4). By using the Chinese Remainder Theorem, we prove that R is isomorphic to a direct sum of four rings. We define a Gray map Phi from R-n to Z(4)(4n), which is a distance preserving map. The Gray image of a cyclic code over R is a linear code over Z(4). We also discuss some properties of MDS codes over R. Furthermore, we study the MacWilliams identities of linear codes over R and give the generator polynomials of cyclic codes over R.

关键词： linear code Gray map MDS codes MacWilliams identities Cyclic codes

来源：评论

学校读者我要写书评

暂无评论

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案：

请选择收藏分类：

通借通还

建议与咨询 留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案： 新增检索档案 确定 取消

请选择收藏分类： 新增自定义分类 确定 取消

通借通还

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

请选择保存的检索档案：

请选择收藏分类：