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Characterizations for minimal codes: graph theory approach and algebraic approach over finite chain rings

Designs, codes, and Cryptography 2025年 1-33页

作者： Maji, Makhan Mesnager, Sihem Sarkar, Santanu Hansda, Kalyan Department of Mathematics Indian Institute of Technology Madras Tamil Nadu Chennai 600036 India Department of Mathematics University of Paris VIII Saint-Denis 93526 France Laboratoire Analyse Géométrie et Applications (LAGA) UMR 7539 CNRS Villetaneuse 93430 France Telecom Paris Palaiseau Paris 91120 France Department of Mathematics Visva-Bharati University West Bengal Santiniketan 731235 India

The concept of minimal linear codes was introduced by Ashikhmin and Barg in 1998, leading to the development of various methods for constructing these codes over finite fields. In this context, minimality is defined as a codeword u in a linear code C is considered minimal if u covers the codeword cu for all c in the finite field Fq of order q but no other codewords in C. A linear code C is said to be minimal if each of its codewords is minimal. minimal codewords are widely used in decoding linear codes, secret sharing schemes, secure two-party computations, cryptography, and other areas such as combinatorics. They have also facilitated the exploration of codes and research codes over finite commutative rings, which are considered appropriate alphabets for coding theory. Extending the minimality property from finite fields to rings and developing such codes poses significant challenges but presents opportunities for advancing coding theory in the context of finite rings. Firstly, the aim is to create graphs that produce a linear minimal (or nearly minimal) code through their adjacency, and examples will be offered for explicit illustrations. Secondly, there is an investigation of codes over rings generated by minimal codewords and an exploration of related minimal codes over finite chain rings. More specifically, a basis C is constructed so that every codeword is minimal. To this end, a linear transformation of C with this basis is built, and sufficient and necessary minimal linear codes over finite chain rings are provided. Then, there is a new design of minimality conditions over finite principal ideal rings. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.

关键词： (Directed) graph Almost-minimal code Commutative ring Cyclic code Finite field Linear code minimal code

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Construction of minimal binary linear codes with dimension n+3

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2025年第2期17卷 433-452页

作者： Shaikh, Wajid M. Jain, Rupali S. Reddy, B. Surendranath Patil, Bhagyashri S. SRTMU Nanded Sch Computat Sci Nanded India

In this paper, for n >= 6, we present the generic construction of binary linear codes of length 2(n) - 1 with dimension n + 3, and derive the necessary and sufficient condition for the constructed codes to be minimal. Using this generic construction, a new family of minimal binary linear codes violating the Ashikhmin-Barg condition will be constructed from a special class of Boolean functions. We also obtain the weight distribution of the constructed minimal binary linear codes. We will achieve minimal codes with the highest dimension, resulting in a better rate of transmission.

关键词： Linear code minimal code Weight distribution Ashikhmin-Barg condition

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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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On the number of minimal codewords in codes generated by the adjacency matrix of a graph

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DISCRETE APPLIED MATHEMATICS 2022年 309卷 221-228页

作者： Kurz, Sascha Univ Bayreuth Fak Math Phys & Informat Bayreuth Germany

minimal codewords have applications in decoding linear codes and in cryptography. We study the number of minimal codewords in binary linear codes that arise by appending a unit matrix to the adjacency matrix of a grap... 详细信息

关键词： minimal code Graphs Linear codes

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Short minimal codes and Covering codes via Strong Blocking Sets in Projective Spaces

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IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第2期68卷 881-890页

作者： Heger, Tamas Nagy, Zoltan Lorant Eotvos Lorand Univ ELKH ELTE Geometr & Algebra Combinator Res Grp H-1117 Budapest Hungary

minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. minimal linear codes have been studied since decades but their tight connection with cutting blocking sets of finite projective spaces was unfolded only in the past few years, and it has not been fully exploited yet. In this paper we apply finite geometric and probabilistic arguments to contribute to the field of minimal codes. We prove an upper bound on the minimal length of minimal codes of dimension k over the q-element Galois field which is linear in both q and k, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well.

关键词： Linear code minimal code covering code saturating set cutting blocking set higgledy-piggledy line set

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THREE COMBINATORIAL PERSPECTIVES ON minimal codeS

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SIAM JOURNAL ON DISCRETE MATHEMATICS 2022年第1期36卷 461-489页

作者： Alfarano, Gianira N. Borello, Martino Neri, Alessandro Ravagnani, Alberto Univ Zurich Inst Math CH-8057 Zurich Switzerland Univ Sorbonne Paris Nord CNRS UMR 7539 Lab Geometrie Anal & ApplicatLAGAUniv Paris 8 Paris France Max Planck Inst Math Sci Inselstr 22 D-04103 Leipzig Germany Eindhoven Univ Technol Dept Math & Comp Sci Eindhoven Netherlands

We develop three approaches of combinatorial flavor to study the structure of minimal codes and cutting blocking sets in finite geometry, each of which has a particular application. The first approach uses techniques from algebraic combinatorics, describing the supports in a linear code via the Alon-Ftiredi theorem and the combinatorial Nullstellensatz. The second approach combines methods from coding theory and statistics to compare the mean and variance of the nonzero weights in a minimal code. Finally, the third approach regards minimal codes as cutting blocking sets and studies these using the theory of spreads in finite geometry. By applying and combining these approaches with each other, we derive several new bounds and constraints on the parameters of minimal codes. Moreover, we obtain two new constructions of cutting blocking sets of small cardinality in finite projective spaces. In turn, these allow us to give explicit constructions of minimal codes having short length for the given field and dimension.

关键词： minimal code cutting blocking set strong blocking set spread combinatorial Nullstellensatz Pless identity

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On the weight distribution of some minimal codes

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DESIGNS codeS AND CRYPTOGRAPHY 2021年第3期89卷 471-487页

作者： Bartoli, Daniele Bonini, Matteo Timpanella, Marco Univ Perugia Dipartimento Matemat & Informat Perugia Italy Univ Trento Dipartimento Matemat Trento Italy Univ Basilicata Dipartimento Matemat Informat & Econ Potenza Italy

minimal codes are a class of linear codes which gained interest in the last years, thanks to their connections to secret sharing schemes. In this paper we provide the weight distributions and the parameters of familie... 详细信息

关键词： Linear code minimal code Weight distribution

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A Novel Application of Boolean Functions With High Algebraic Immunity in minimal codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2021年第10期67卷 6856-6867页

作者： Chen, Hang Ding, Cunsheng Mesnager, Sihem Tang, Chunming China West Normal Univ Sch Math & Informat Nanchong 637002 Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China Univ Paris VIII Dept Math F-93526 St Denis France Univ Sorbonne Paris Cite Lab Anal Geometry & Applicat LAGA CNRS UMR 7539 F-93430 Villetaneuse France Telecom Paris F-91120 Palaiseau France

Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing minimal binary codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity, are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension m and length less than or equal to m(m + 1)/2 are obtained. Besides, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields additionally to symmetric cryptography, such as coding theory and secret sharing schemes.

关键词： Boolean function vectorial Boolean function algebraic immunity linear code Reed-Muller code minimal code secret sharing

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On One-Dimensional Linear minimal codes Over Finite (Commutative) Rings

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IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第5期68卷 2990-2998页

作者： Maji, Makhan Mesnager, Sihem Sarkar, Santanu Hansda, Kalyan Visva Bharati Univ Dept Math Bolpur 731235 W Bengal India Univ Paris VIII Dept Math F-93526 St Denis France CNRS Lab Anal Geometrie & Applicat LAGA UMR 7539 F-93430 Villetaneuse France IIT Madras Dept Math Chennai 600036 Tamil Nadu India

minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. When they are defined over finite fields, those codes have been intensively studied, especially in recent years, but they have been firstly partially characterized by Ashikhmin and Barg since 1998. Next, they were completely characterized in 2018 by Ding, Heng, and Zhou in terms of the minimum and maximum nonzero weights in the corresponding codes. Since then, many construction methods for minimal linear codes over finite fields throughout algebraic and geometric approaches have been proposed in the literature. In particular, the algebraic approach gives rise to minimal codes from (cryptographic) functions. Linear codes over finite fields have been expanded into the collection of acceptable alphabets for codes and study codes over finite commutative rings. A natural way to extend the known results available in the literature is to consider minimal linear codes over commutative rings with unity. In extending coding theory to codes over rings, several essential principles must be considered. Particularly extending the minimality property from finite fields to rings and creating such codes is not simple. Such an extension offers more flexibility in the construction of minimal codes. The present article investigates one-dimensional minimal linear codes over the rings Z(pn) (where p is a prime) and Z(pmqn) (where p < q are distinct primes and m <= n). Our ultimate objective is to characterize such codes' minimality and design minimal linear codes over the considered rings. Given our objective, we first introduced the notion of minimal codes over (commutative) rings and succeeded in deriving simple characterization of one-dimensional minimal linear codes over the underlying rings mentioned above. Our new algebraic approach allows designing new minimal linear codes. Almost minimal codes over rings are also presented. To the best of our knowledge, the present paper

关键词： Algebra coding theory commutative ring linear code minimal code

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minimal Binary codes over a Ring

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湖州师范学院学报 2025年第2期47卷 1-7页

作者： PAN Yang GUO Yuxin YE Chang Wuxi Institute of Technology Wuxi 214121China School of Science Huzhou UniversityHuzhou 313000China

We construct an infinite family of minimal linear codes over the ring F_(2)+u F_(2).These codes are defined through trace functions and Boolean *** Lee weight distribution is completely computed by Walsh *** Gray mapping,we obtain a family of minimal binary linear codes from a generic construction,which have prominent applications in secret sharing and secure two-party computation.

关键词： Boolean function trace function minimal code secret sharing

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