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On a class of optimal constant weight ternary codes

DESIGNS codeS AND CRYPTOGRAPHY 2023年第1期91卷 45-54页

作者： Kharaghani, Hadi Suda, Sho Zaitsev, Vlad Univ Lethbridge Dept Math & Comp Sci Lethbridge AB T1K 3M4 Canada Natl Def Acad Japan Dept Math Yokosuka Kanagawa 2398686 Japan

A weighing matrix W of order n = p(m+1)-1/p-1 and weight p(m) is constructed and shown that the rows of W and -W together form optimal constant weight ternary codes of length n, weight p(m) and minimum distance p(m-1) (p+3/2) for each odd prime power p and integer m >= 1 and thus A(3) (p(m+1)-1/p - 1, p(m-1)(p + 3/2), p(m)) = 2(p(m+1)-1/p - 1).

关键词： Weighing matrix ternary code Constant weight code Optimal code Johnson bound

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Reconfigurable mechanism-based metamaterials for ternary-coded elastic wave polarizers and programmable refraction control

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Applied Mathematics and Mechanics(English Edition) 2024年第7期45卷 1225-1242页

作者： Zhou HU Zhibo WEI Yan CHEN Rui ZHU School of Aerospace Engineering Beijing Institute of TechnologyBeijing100081China School of Mechanical Engineering Tianjin UniversityTianjin300350China Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education Tianjin UniversityTianjin300350China

Elastic metamaterials with unusual elastic properties offer unprecedented ways to modulate the polarization and propagation of elastic ***,most of them rely on the resonant structural components,and thus are frequency-dependent and ***,we present a reconfigurable 2D mechanism-based metamaterial which possesses transformable and frequency-independent elastic *** on the proposed mechanism-based metamaterial,interesting functionalities,such as ternarycoded elastic wave polarizer and programmable refraction,are ***,unique ternary-coded polarizers,with 1-trit polarization filtering and 2-trit polarization separating of longitudinal and transverse waves,are first ***,the strong anisotropy of the proposed metamaterial is harnessed to realize positive-negative bi-refraction,only-positive refraction,and only-negative ***,the wave functions with detailed microstructures are numerically verified.

关键词： elastic metamaterial elastic wave reconfigurable design zero mode ternary code programmable refraction

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On some conjectures about optimal ternary cyclic codes

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2022年第4期33卷 419-436页

作者： Liu, Qian Liu, Ximeng Fuzhou Univ Coll Math & Comp Sci Fuzhou 350108 Peoples R China Fuzhou Univ Key Lab Informat Secur Network Syst Fuzhou 350108 Peoples R China

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, by investigating the solutions of certain equations over finite fields, we make progress towards three conjectures about optimal ternary cyclic codes which were proposed by Ding and Helleseth.

关键词： Cyclic code Optimal code ternary code Sphere packing bound

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Linear codes of 2-designs as subcodes of the generalized Reed-Muller codes

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2021年第3期13卷 407-423页

作者： He, Zhiwen Wen, Jiejing Zhejiang Univ Sch Math Sci Hangzhou 310027 Peoples R China Shandong Univ Key Lab Cryptol Technol & Informat Secur Minist Educ Qingdao 266237 Peoples R China Shandong Univ Sch Cyber Sci & Technol Qingdao 266237 Peoples R China

This paper is devoted to the affine-invariant ternary codes defined by Hermitian functions. We first compute the incidence matrices of the 2-designs supported by the minimum weight codewords of these ternary codes. Then we show that the linear codes spanned by the rows of these incidence matrices are subcodes of the 4-th order generalized Reed-Muller codes and also hold 2-designs. Finally, we determine the dimension and develop a lower bound on the minimum distance of the ternary linear codes.

关键词： ternary code 2-design Incidence matrix Generalized Reed-Muller code

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Characterization and classification of optimal LCD codes

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DESIGNS codeS AND CRYPTOGRAPHY 2021年第4期89卷 617-640页

作者： Araya, Makoto Harada, Masaaki Saito, Ken Shizuoka Univ Dept Comp Sci Hamamatsu Shizuoka 4328011 Japan Tohoku Univ Res Ctr Pure & Appl Math Grad Sch Informat Sci Sendai Miyagi 9808579 Japan

Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. We give a characterization of LCD codes over F-q having large minimum weights for q is an element of{2,3}. Using the characterization, for arbitrary n, we determine the largest minimum weights among LCD [n, k] codes over F-q, where (q, k)is an element of{(2,4),(3,2),(3,3)}. Moreover, for arbitrary n, we give a complete classification of optimal LCD [n, k] codes over F-q, where (q, k) is an element of{(2,3),(2,4),(3,2),(3,3)}.

关键词： Linear complementary dual code Binary code ternary code Simple code Griesmer bound

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On the minimum weights of binary LCD codes and ternary LCD codes

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FINITE FIELDS AND THEIR APPLICATIONS 2021年 76卷 101925-101925页

作者： Araya, Makoto Harada, Masaaki Saito, Ken Shizuoka Univ Dept Comp Sci Hamamatsu Shizuoka 4328011 Japan Tohoku Univ Grad Sch Informat Sci Res Ctr Pure & Appl Math Sendai Miyagi 9808579 Japan

Linear complementary dual (LCD) codes are linear codes that intersect with their dual codes trivially. We study the largest minimum weight d(2)(n, k) among all binary LCD [n, k] codes and the largest minimum weight d(3)(n, k) among all ternary LCD [n, k] codes. The largest minimum weights d(2)(n, 5) and d(3)(n,4) are partially determined. We also determine the largest minimum weights d(2)(n, n - 5), d(3)(n, n - i) for i is an element of {2, 3, 4}, and d(3)(n, k) for n is an element of {11, 12, ..., 19}. (C) 2021 Elsevier Inc. All rights reserved.

关键词： Linear complementary dual code Binary code ternary code Minimum weight

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On an open problem about a class of optimal ternary cyclic codes

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FINITE FIELDS AND THEIR APPLICATIONS 2019年 59卷 335-343页

作者： Han, Dongchun Yan, Haode Southwest Jiaotong Univ Sch Math Chengdu 610031 Sichuan Peoples R China

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth [5]. Let C-(1,C-e) be a cyclic code of length 3(m) - 1 over GF(3) with two nonzeros alpha and alpha(e), where m and e are given integers, and alpha is a generator of GF(3(m))*. It is shown that C-(1,C-e) is optimal with parameters [3(m) - 1, 3(m)-1-2m, 4] if one of the following conditions is met. 1) m equivalent to 0(mod 4), m >= 4, and e = 3(m/2) + 5.2) m equivalent to 2(mod 4), m >= 6, and e = 3(m+2/2) + 5. (C) 2019 Published by Elsevier Inc.

关键词： Cyclic code Optimal code ternary code Sphere Packing bound

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Four-Mode Programmable Metamaterial Using ternary Foldable Origami

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ACS APPLIED MATERIALS & INTERFACES 2019年第31期11卷 28554-28561页

作者： Dinh Hai Le Lim, Sungjoon Chung Ang Univ Sch Elect & Elect Engn 221 Heukseok Dong Seoul 06974 South Korea

Designing a multifunctional metamaterial with programmable feature has become a new trend in mechanical, acoustic, and electromagnetic research fields due to the controllability of their structural behaviors and functionalities. The codable or reconfigurable structures have shown more remarkable characteristics than the traditional and conventional metamaterials to implement functional programmability. However, structural complexity and hi-tech requirement are the biggest constraints to their practical applications. This paper numerically and experimentally investigates a programmable metamaterial based on ternary foldable origami in the gigahertz-frequency regime. The proposed metamaterial provides four transformable modes corresponding to four different functions of electromagnetic reflector and frequency-selectable absorbers by programming unique ternary foldable origami coded as "0", "1", and "2" for different folding levels. Interestingly, the proposed foldable origami consists of a simple dielectric paper and a bottom conductor, while there is no conductive pattern on the top. Therefore, the proposed programmable metamaterial is extremely robust and can be extended to a multiresonance mode and origami computing.

关键词： metamaterial programmable metamaterial coding metamaterial ternary code origami

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A family of optimal ternary cyclic codes from the Niho-type exponent

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FINITE FIELDS AND THEIR APPLICATIONS 2018年第Nov.期54卷 101-112页

作者： Yan, Haode Zhou, Zhengchun Du, Xiaoni Southwest Jiaotong Univ Sch Math Chengdu 610031 Sichuan Peoples R China State Key Lab Cryptol POB 5159 Beijing 100878 Peoples R China Northwest Normal Univ Coll Math & Stat Lanzhou 730070 Gansu Peoples R China

Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. The objective of this paper is to present a family of optimal ternary cyclic codes from the Niho-type exponent. Specifically, for an odd integer m and a positive integer r with 4r 1(mod m), a family of cyclic codes C-(u,C- v) of length 3(m) - 1 over GF(3) with two nonzeros beta(u) and beta(v) is studied, where beta is a generator of GF(3(m))*, u = (3(m) + 1)/2 and v = 3(r) + 2 is the ternary Niho-type exponent. The parameters of this family of cyclic codes are determined. It turns out that C-(u,C- v) is optimal with respect to the Sphere Packing bound if 9 (sic) m and otherwise almost optimal. Thanks to a recent proof of the Dobbertin-Helleseth-Kumar-Martin conjecture by Katz and Langevin, the dual of this family of cyclic codes is shown to have at most five nonzero weights. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Cyclic code Optimal code ternary code Sphere Packing bound Niho exponent

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On the classification of certain ternary codes of length 12

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HIROSHIMA MATHEMATICAL JOURNAL 2016年第1期46卷 87-96页

作者： Araya, Makoto Harada, Masaaki Shizuoka Univ Fac Informat Dept Comp Sci Hamamatsu Shizuoka 4328011 Japan Tohoku Univ Grad Sch Informat Sci Res Ctr Pure & Appl Math Sendai Miyagi 9808579 Japan

Shimada and Zhang studied the existence of polarizations on some super-singular K3 surfaces by reducing the existence of the polarizations to that of ternary [12, 5] codes satisfying certain conditions. In this note, we give a classification of ternary [12, 5] codes satisfying the conditions. To do this, ternary [10, 5] codes are classified for minimum weights 3 and 4.

关键词： ternary code classification weight enumerator

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