检索结果-内蒙古大学图书馆

Homogenization of binary linear codes and their applications

FINITE FIELDS AND THEIR APPLICATIONS 2025年 103卷

作者： Hyun, Jong Yoon Mondal, Nilay Kumar Lee, Yoonjin Konkuk Univ Glocal Campus268 Chungwon Daero Chungju Si 27478 Chungcheongbuk South Korea Ewha Womans Univ Inst Math Sci Seoul 03760 South Korea Ewha Womans Univ Dept Math Seoul 03760 South Korea

We introduce a new technique, called homogenization, for a systematic construction of augmented codes of binary linear codes, using the defining set approach in connection to multivariable functions. We explicitly determine the parameters and the weight distribution of the homogenized codes when the defining set is either a simplicial complex generated by any finite number of elements, or the difference of two simplicial complexes, each of which is generated by a single maximal element. Using this homogenization technique, we produce several infinite families of optimal codes, self-orthogonal codes, minimal codes, and self-complementary codes. As applications, we obtain some best known quantum errorcorrecting codes, infinite families of intersecting codes (used in the construction of covering arrays), and we compute the Trellis complexity (required for decoding) for several families of codes as well. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Homogenization Multi-variable function Simplicial complex optimal code Binary code

来源：评论

学校读者我要写书评

暂无评论

On optimal Finite-Length Block codes of Size Four for Binary Symmetric Channels

引用

IEEE TRANSACTIONS ON INFORMATION THEORY 2025年第1期71卷 138-166页

作者： Dong, Yanyan Yang, Shenghao Chinese Univ Hong Kong Sch Sci & Engn Shenzhen 518172 Guangdong Peoples R China

An (n,M) code refers to a binary code with blocklength n and codebook size M. Such codes are studied in the context of memoryless binary symmetric channels (BSCs) with maximum likelihood (ML) decoding. Previous research has characterized some optimal codes among the linear (n,4) codes for any n >= 2 . However, it was unknown whether these optimal codes among linear codes were better than all nonlinear codes. In this paper, we first demonstrate that for any n >= 2 , there exists an optimal code among all (n,4) codes that is either linear or belongs to a subset of nonlinear codes called Class-I codes. We identify all the optimal codes among the linear (n,4) codes for each blocklength n >= 2 and discover some that were not previously reported in the literature. For any n from 2 to 8, all the optimal (n,4) codes are identified. Except for n = 3 , all the optimal (n,4) codes are equivalent to linear codes. There exist optimal (3,4) codes that are not equivalent to linear codes. Furthermore, we introduce a subset of nonlinear codes called Class-II codes and show that for any n > 3 , the set composed of linear, Class-I, and Class-II codes and their equivalent codes contains all the optimal (n,4) codes. Both Class-I and Class-II codes are close to linear codes in the sense that they involve only one type of column that is not included in linear codes. We derive a sufficient condition such that all the optimal (n,4) codes are equivalent to linear codes, which can be evaluated by computer with a computation cost O(n(6)) .

关键词： codes Linear codes Maximum likelihood decoding Costs Computers Binary codes Vectors Sufficient conditions Parity check codes Hamming distances Binary symmetric channel finite blocklength optimal code

来源：评论

学校读者我要写书评

暂无评论

Three new classes of optimal quinary cyclic codes with minimum distance four

引用

APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2025年第3期36卷 493-501页

作者： Liu, Yan Cao, Xiwang Yancheng Inst Technol Coll Math & Phys Yancheng 224003 Peoples R China Nanjing Univ Aeronaut & Astronaut Sch Math Sci Nanjing 210016 Peoples R China MIIT Key Lab Math Modeling & High Performance Comp Air Nanjing 210016 Peoples R China

Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an important subject of study for many years. Recently, several classes of optimal quinary cyclic codes of the forms C(0,1,e) and C(1,e,s) are presented in the literature, where s = 5(m)-1/(2) and 2 = e = 5(m) - 2. In this paper, by considering the solutions of certain equations over finite fields, we give three new classes of infinite families of optimal quinary cyclic codes of the form C(1,e,s) with parameters [5m - 1, 5(m) - 2(m) - 2, 4] . Specifically, we make progress towards an open problem proposed by Gaofei Wu et al. [17].

关键词： Cyclic code optimal code Quinary code Minimum distance

来源：评论

学校读者我要写书评

暂无评论

Minimal and optimal binary codes obtained using CD-construction over the non-unital ring I

引用

DESIGNS codeS AND CRYPTOGRAPHY 2024年第1期92卷 145-157页

作者： Sagar, Vidya Sarma, Ritumoni Indian Inst Technol Delhi Dept Math Delhi 110016 India

In this article, we construct linear codes over the commutative non-unital ring I of size four. We obtain their Lee-weight distributions and study their binary Gray images. Under certain mild conditions, these classes of binary codes are minimal and self-orthogonal. All codes in this article are few-weight codes. Besides, an infinite class of these binary codes consists of distance optimal codes with respect to the Griesmer bound.

关键词： Few-weight code Minimal code optimal code Self-orthogonal code Non-unital ring Simplicial complex

来源：评论

学校读者我要写书评

暂无评论

Explicit Cyclic and Quasi-Cyclic codes With optimal, Best Known Parameters, and Large Relative Minimum Distances

引用

IEEE TRANSACTIONS ON INFORMATION THEORY 2024年第12期70卷 8688-8697页

作者： Xie, Conghui Chen, Hao Yuan, Chen Jinan Univ Coll Informat Sci & Technol Cyber Secur Guangzhou 510632 Guangdong Peoples R China Shanghai Jiao Tong Univ Sch Elect Informat & Elect Engn Shanghai 200240 Peoples R China

In this paper, we construct many infinite families of distance-optimal codes with new parameters, some of which are BCH codes and quasi-cyclic codes. In particular, we report the first infinite family of binary distance-optimal BCH codes with the minimum distance 8. Secondly, several infinite families of binary BCH codes and quasi-cyclic codes are presented. Many codes in these families have optimal or best known parameters. Thirdly, we construct infinite families of binary cyclic [n,>= n+1/2, d](2) codes with minimum distances d >= [n-1/Pi(s)(i =1) p(i)], n = (2(p1) - 1)(2(p2) - 1) center dot center dot center dot (2(ps) - 1), p(1),..., p(s) are different primes. Our construction extends the main result of a recent paper published by Sun et al. to much more general binary cyclic codes with various lengths. We also construct an infinite family of binary quasi-cyclic codes with the rate around 1 2 and relative minimum distance lower bounded by O( 1/log(2) log(2) n).

关键词： Distance-optimal code binary BCH code optimal code best known code cyclic code quasi-cyclic code relative minimum distance

来源：评论

学校读者我要写书评

暂无评论

Infinite Families of Few Weight optimal Binary Linear codes From Multivariable Functions

引用

IEEE TRANSACTIONS ON INFORMATION THEORY 2024年第10期70卷 6912-6931页

作者： Hyun, Jong Yoon Jeong, Jihye Lee, Yoonjin Konkuk Univ Coll Liberal Arts Glocal Campus Chungju Si 27478 Chungcheongbuk South Korea Ewha Womans Univ Dept Math Seoul 03760 South Korea

We study the binary linear code families associated with certain types of multivariable functions. We observe that a majority of these codes are not optimal codes nor few weight codes yet. In this paper, we find infinite families of few weight (near-) optimal binary linear codes from our code families. Furthermore, we produce support t-designs ( t = 2 or 3) which cannot be determined by the Assmus-Mattson Theorem;this is the first time that the result by Tang et al. was successfully used to prove that linear codes hold t-designs. As another application, we find many (near-) optimal quantum codes from the dual codes of our code families using the CSS construction. As a main method, we use the modified shortening method (simply, called shortening method), which is applied to our code families. Using the results on the weight distributions of our shortened codes, we verify that our codes families support t-designs ( t = 2, 3 ). We emphasize that some infinite families of few weight optimal binary linear codes have new parameters.

关键词： optimal code few weight code multivariable function t-design quantum code optimal code few weight code multivariable function t-design quantum code

来源：评论

学校读者我要写书评

暂无评论

On optimal constant weight codes derived from ω-circulant balanced generalized weighing matrices

引用

DESIGNS codeS AND CRYPTOGRAPHY 2024年第10期92卷 2791-2799页

作者： Kharaghani, Hadi Pender, Thomas Tonchev, Vladimir Univ Lethbridge Dept Math & Comp Sci Lethbridge AB T1K 3M4 Canada Univ Simon Fraser Dept Math Burnaby BC V5A 1S6 Canada Michigan Technol Univ Math Sci Houghton MI USA

Balanced generalized weight matrices are used to construct optimal constant weight codes that are monomially inequivalent to codes derived from the classical simplex codes. What's more, these codes can be assumed to be generated entirely by omega-shifts of a single codeword where omega is a primitive element of a Galois field. Additional constant weight codes are derived by projecting onto subgroups of the alphabet sets. These too are shown to be optimal.

关键词： Constant weight code Equidistant nonlinear code Inequivalent codes optimal code omega-circulant balanced generalized weighing matrix

来源：评论

学校读者我要写书评

暂无评论

The Griesmer codes of Belov type and optimal quaternary codes via multi-variable functions

引用

CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2024年第3期16卷 579-600页

作者： Hyun, Jong Yoon Han, Nayoung Lee, Yoonjin Konkuk Univ 268 Chungwon-daeroGlocal Campus Chungju 27478 South Korea Ewha Womans Univ Dept Math 52 Ewhayeodae gil Seoul 03760 South Korea

We study the Griesmer codes of specific Belov type and construct families of distance-optimal linear codes over Z(4) by using multi-variable functions. We first show that the pre-images of specific Griesmer codes of Belov type under a Gray map phi from Z(4) to Z(2)(2) are non-linear except one case. Therefore, we are interested in finding subcodes of Griesmer codes of specific Belov type with maximum possible dimension whose pre-images under phi are still linear over Z(4) such that they also have good properties such as optimality and two-weight. To this end, we introduce a new approach for constructing linear codes over Z4 using multi-variable functions over Z. This approach has an advantage in explicitly computing the Lee weight enumerator of a linear code over Z(4). Furthermore, we obtain several other families of distance-optimal two-weight linear codes over Z(4) by using a variety of multi-variable functions. We point out that some of our families of distance-optimal codes over Z(4) have linear binary Gray images which are also distance-optimal.

关键词： optimal code Griesmer code Belov type Few-weight code Gray map

来源：评论

学校读者我要写书评

暂无评论

optimal quinary cyclic codes with three zeros

引用

FINITE FIELDS AND THEIR APPLICATIONS 2025年 101卷

作者： Fan, Jinmei Zeng, Xiangyong Guilin Univ Technol Sch Math & Stat Guilin 541004 Peoples R China Hubei Univ Minist Educ Key Lab Intelligent Sensing Syst & Secur Fac Math & StatHubei Key Lab Appl Math Wuhan 430062 Peoples R China

optimal cyclic codes have received a lot of attention and much progress has been made. However, little is known about optimal quinary cyclic codes. In this paper, by analyzing irreducible factors of certain polynomials over finite fields and utilizing multivariate method, three classes of optimal quinary cyclic codes with parameters [5m -1, 5m -2m -2,4] and three zeros are presented. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Cyclic code optimal code Irreducible polynomial Finite field

来源：评论

学校读者我要写书评

暂无评论

optimal Few-SSW Linear codes and Their Subcode Support Weight Distributions

引用

IEEE TRANSACTIONS ON INFORMATION THEORY 2025年第2期71卷 1028-1042页

作者： Pan, Xu Chen, Hao Liu, Hongwei Liu, Shengwei Jinan Univ Coll Informat Sci & Technol Cyber Secur Guangzhou 510632 Guangdong Peoples R China Jinan Univ Coll Informat Sci & Technol Guangzhou 510632 Guangdong Peoples R China Cent China Normal Univ Sch Math & Stat Key Lab NAA MOE Wuhan 430079 Peoples R China Cent China Normal Univ Sch Math & Stat Wuhan 430079 Peoples R China

Few-weight codes have been constructed and studied for many years, since their fascinating relations to finite geometries, strongly regular graphs and Boolean functions. Simplex codes are one-weight [q(k-1)/q-1, k ,q(k-1)]q -linear codes and they meet all Griesmer bounds on the generalized Hamming weights of linear codes. All the subcodes with dimension r of a [q(k-1)/q-1, k ,q(k-1)]q -simplex code have the same subcode support weight q(k-r) (q(r-1))/q-1 for1 <= r <= k. In this paper, we construct linear codes meeting the Griesmer bound of the r-generalized Hamming weight, such codes do not meet the Griesmer bound of the j-generalized Hamming weight for 1 <= j < r. . Moreover these codes have only few subcode support weights (few-SSW). The weight distributions and the subcode support weight distributions of these distance-optimal codes are determined. Linear codes constructed in this paper are natural generalizations of distance-optimal few-weight codes.

关键词： codes Linear codes Hamming weight Vectors Information science Hamming distances Boolean functions Symbols Standards Reed-Muller codes optimal code r-Griesmer code few subcode support weight code weight distribution subcode support weight distribution

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：