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EFFICIENT GENERATION OF optimal PREFIX code - EQUIPROBABLE WORDS USING UNEQUAL COST LETTERS

JOURNAL OF THE ACM 1975年第2期22卷 202-214页

作者： PERL, Y GAREY, MR EVEN, S WEIZMANN INST SCI REHOVOTHISRAEL BELL TEL LABS INC MURRAY HILLNJ 07974 TECHNION DEPT COMP SCIHAIFAISRAEL

An algorithm for constructing an optimal prefix code of n eqmprobable words over r unequal cost coding letters is given. The discussion is in terms of rooted labeled trees. The algorithm consists of two parts. The first one is an extension algorithm which constructs a prefix code of n words. This code is either optimal or is a “good” approximation The second part is a mending algorithm which changes the code constructed by the extension algorithm into an optimal code in case it is not already optimal. The validity of the combined algorithm is proved and its structure is analyzed. The analysis leads to further improvement of the algorithm's efficiency. It is shown that the number of steps required is at mnst O(*** n), if a heap data structure is used Alternatively, one can use a data structure of r queues, in which case the number of steps is bounded by O(r.n). © 1975, ACM. All rights reserved.

关键词： equiprobable code words minimum redundancy code optimal code prefix code tree

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Homogenization of binary linear codes and their applications

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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

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

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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

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

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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

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optimal quinary cyclic codes with three zeros

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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

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

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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

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Constructing optimal few weight quaternary linear codes via multivariable functions

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2025年第1期17卷 57-85页

作者： Hyun, Jong Yoon Jeong, Jihye Lee, Yoonjin Konkuk Univ Glocal Campus268Chungwon Daero Chungjusi 27478 Chungcheongbug South Korea Ewha Womans Univ Dept Math 52Ewhayeodae Gil Seoul 03760 South Korea

We construct new infinite families of (near) Plotkin-optimal 2-weight or 3-weight codes over Z4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_4$$\end{document} by using the shortening method. Furthermore, we completely determine the Lee weight distributions of our shortened codes. To achieve our goal, we use certain families of multivariable functions, and we interpret a shortening method followed by puncturing in terms of multivariable functions. According to this interpretation, we find explicit criteria for the shortened codes to have fewer Lee weights and larger minimum Lee weights after the shortening process. As our contribution, we emphasize that non-Plotkin-optimal code families are converted to Plotkin-optimal code families after the shortening process by using our main results. Furthermore, we produce new infinite families of (near) Plotkin-optimal 2-weight or 3-weight codes over Z4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_4$$\end{document}, which extend the database of linear codes over Z4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_4$$\end{document}.

关键词： Shortening optimal code Quaternary code Few weight Multivariable function

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Subfield codes of CD-codes over F2[x]/⟨x³ - x⟩

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DISCRETE MATHEMATICS 2025年第1期348卷

作者： Bhagat, Anuj Kumar Sarma, Ritumoni Sagar, Vidya Indian Inst Technol Delhi Dept Math New Delhi 110016 India

A non-zero F-linear map from a finite-dimensional commutative F-algebra to the field Fis called an F-valued trace if its kernel does not contain any non-zero ideals. In this article, we utilize an F2-valued trace of the F2-algebra R2:= F2[x]/x3- x to study binary subfield code C(2) Dof CD:={(x center dot d) d.D: x. Rm2} for each defining set Dderived from a certain simplicial complex. For m. Nand X.{1, 2,..., m}, define X:={v. Fm2: Supp(v). X} and D :=(1 + u2) D1+ u2D2+(u + u2)D3, a subset of Rm2, where u = x + x3- x, D1.{L, cL}, D2.{ M, cM} and D3.{N, cN}, for L, M, N.{1, 2,..., m}. The parameters and the Hamming weight distribution of the binary subfield code C(2)Dof CDare determined for each D. These binary subfield codes are minimal under certain mild conditions on the cardinalities of L, Mand N. Moreover, most of these codes are distanceoptimal. Consequently, we obtain a few infinite families of minimal, self-orthogonal and distance-optimal binary linear codes that are either 2-weight or 4-weight. It is worth mentioning that we have obtained several new distance-optimal binary linear codes. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Linear code Subfield code Minimal code optimal code Self-orthogonal code Simplicial complex

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Families of optimal codes for strong identification

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DISCRETE APPLIED MATHEMATICS 2002年第1-3期121卷 203-213页

作者： Laihonen, T Ranto, S Univ Turku Dept Math FIN-20014 Turku Finland Univ Turku Turku Ctr Comp Sci FIN-20014 Turku Finland

codes for strong identification are considered. The motivation for these codes comes from locating faulty processors in a multiprocessor system, Constructions and lower bounds on these codes are given. In particular, we provide two infinite families of optimal strongly identifying codes, which can locate up to two malfunctioning processors in a binary hypercube. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： binary code identifying code strongly identifying code optimal code Hamming space

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Searching for (near) optimal codes 9th

Searching for (near) Optimal Codes

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9th Annual International Conference on Combinatorial Optimization and Applications (COCOA)

作者： Li, Xueliang Mao, Yaping Wei, Meiqin Li, Ruihu Nankai Univ Ctr Combinator Tianjin 300071 Peoples R China Nankai Univ LPMC TJKLC Tianjin 300071 Peoples R China Qinghai Normal Univ Dept Math Xining 810008 Qinghai Peoples R China Air Force Engn Univ Inst Sci Xian 710051 Peoples R China

ISBN: (纸本)9783319266268;9783319266251

Formally self-dual (FSD) codes are interesting codes and have received an enormous research effort due to their importance in mathematics and computer science. Danielsen and Parker proved that every self-dual additive code over GF(4) is equivalent to a graph codes in 2006, and hence graph is an important tool for searching (near) optimal codes. In this paper, we introduce a new method of searching (near) optimal binary (formally self-dual) linear codes and additive codes from circulant graphs.

关键词： Graph code FSD code Additive code optimal code Circulant graph

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