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Hulls of Reed-Solomon codes via algebraic geometry codes

IEEE TRANSACTIONS ON INFORMATION THEORY 2023年第2期69卷 1005-1014页

作者： Chen, Bocong Ling, San Liu, Hongwei South China Univ Technol Sch Math Guangzhou 510641 Peoples R China Nanyang Technol Univ Sch Phys & Math Sci Singapore 637371 Singapore Cent China Normal Univ Sch Math & Stat Wuhan 430079 Peoples R China

Let RSk(a) be a k-dimensional Reed-Solomon (RS) code over F-q associated with a = (alpha(1), ..., alpha(n)) and let h = Pi(n)(i=1)(z - alpha(i)) be a polynomial in variable z. In this paper, by expressing RSk(a) as an L-construction algebraic geometry code, we completely determine the dimension of the hull RSk(a) boolean AND RSk(a)(perpendicular to) in terms of the degree of the derivative of h and some relevant polynomials. As applications, we explicitly determine the parameters of MDS entanglement-assisted quantum error-correcting codes constructed from RS codes, and all linear complementary dual (resp. self-dual) RS codes are also fully described.

关键词： Hull of a code Reed-Solomon code algebraic geometry code

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On the Feng-Rao bound for the L-construction of algebraic geometry codes

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2000年第5期E83A卷 923-926页

作者： Matsumoto, R Miura, S Tokyo Inst Technol Dept Elect & Elect Engn Tokyo 1528552 Japan Sony Corp Informat & Network Technol Labs Tokyo 1410001 Japan

We show how to apply the Feng-Rao decoding algorithm and the Feng-Rao bound for the Omega-construction of algebraic geometry codes to the L-construction. Then we give examples in which the L-construction gives better ... 详细信息

关键词： algebraic geometry code minimum distance decoding L-construction

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Non-Binary Polar codes using Reed-Solomon codes and algebraic geometry codes

Non-Binary Polar Codes using Reed-Solomon Codes and Algebrai...

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IEEE Information Theory Workshop (ITW)

作者： Mori, Ryuhei Tanaka, Toshiyuki Kyoto Univ Grad Sch Informat Kyoto 6068501 Japan

ISBN: (纸本)9781424482641

Polar codes, introduced by Arikan, achieve symmetric capacity of any discrete memoryless channels under low encoding and decoding complexity. Recently, non-binary polar codes have been investigated. In this paper, we calculate error probability of non-binary polar codes constructed on the basis of Reed-Solomon matrices by numerical simulations. It is confirmed that 4-ary polar codes have significantly better performance than binary polar codes on binary-input AWGN channel. We also discuss an interpretation of polar codes in terms of algebraic geometry codes, and further show that polar codes using Hermitian codes have asymptotically good performance.

关键词： AWGN channels Hermitian codes Hermitian matrices Reed-Solomon code Reed-Solomon codes algebraic geometric codes algebraic geometry code binary input AWGN channel error probability error statistics nonbinary polar code

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On the Hermitian Hulls of Two-Point algebraic geometry codes

On the Hermitian Hulls of Two-Point Algebraic Geometry Codes

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2024 Information Theory Workshop

作者： Sok, Lin Ezerman, Martianus Frederic Ling, San Nanyang Technol Univ Sch Phys & Math Sci 21 Nanyang Link Singapore Singapore

We study the Hermitian hulls of two-point algebraic geometry codes. Under specific conditions on the Weil differential form associated with the Hermitian dual code, we explicitly determine the hull dimension. We const... 详细信息

ISBN: (纸本)9798350348941;9798350348934

关键词： algebraic curve algebraic geometry code Hermitian hull MDS code self-orthogonal code

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On linear complementary pairs of algebraic geometry codes over finite fields

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DISCRETE MATHEMATICS 2024年第12期347卷

作者： Bhowmick, Sanjit Dalai, Deepak Kumar Mesnager, Sihem Natl Inst Sci Educ & Res Sch Math Sci Bhubaneswar 752050 Odisha India Univ Paris VIII Dept Math F-93526 St Denis France Univ Sorbonne Paris Nord Lab Anal Geometry & Applicat LAGA CNRS UMR 7539 F-93430 Villetaneuse France Inst Polytech Paris Telecom Paris F-91120 Palaiseau France

Linear complementary dual (LCD) codes and linear complementary pairs (LCP) of codes have been proposed for new applications as countermeasures against side-channel attacks (SCA) and fault injection attacks (FIA) in the context of direct sum masking (DSM). The countermeasure against FIA may lead to a vulnerability for SCA when the whole algorithm needs to be masked (in environments like smart cards). This led to a variant of the LCD and LCP problems, where several results were obtained intensively for LCD codes, but only partial results were derived for LCP codes. Given the gap between the thin results and their particular importance, this paper aims to reduce this by further studying the LCP of codes in special code families and, precisely, the characterization and construction mechanism of LCP codes of algebraic geometry codes over finite fields. Notably, we propose constructing explicit LCP of codes from elliptic curves. Besides, we also study the security parameters of the derived LCP of codes (C, D) (notably for cyclic codes), which are given by the minimum distances d(C) and d(D perpendicular to). Further, we show that for LCP algebraic geometry codes (C, D), the dual code C perpendicular to is equivalent to D under some specific conditions we exhibit. Finally, we investigate whether MDS LCP of algebraic geometry codes exist (MDS codes are among the most important in coding theory due to their theoretical significance and practical interests). Construction schemes for obtaining LCD codes from any algebraic curve were given in 2018 by Mesnager, Tang and Qi in [11]. To our knowledge, it is the first time LCP of algebraic geometry codes has been studied. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Finite field Linear complementary pairs (LCP) of codes algebraic geometry code algebraic curve Elliptic curves

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Bounds for generalized Hamming weights of general AG codes

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FINITE FIELDS AND THEIR APPLICATIONS 2015年 34卷 265-279页

作者： Lee, Kwankyu Chosun Univ Dept Math Educ Kwangju 501759 South Korea

We present a good bound for the generalized Hamming weights of multi-point evaluation and differential AG codes. It is a natural generalization of the order bound for one-point AG codes. As an example, we determine the third generalized Hamming weight of one particular two-point evaluation Hermitian code. Then we extend the bound for the relative generalized Hamming weights, which are essential measures of the performance of secret sharing schemes based on linear codes. The access structure of one particular secret sharing scheme based on the previous two-point Hermitian code and its subcode is analyzed using the new bound. (C) 2015 Elsevier Inc. All rights reserved.

关键词： algebraic geometry code Generalized Hamming weights Relative generalized Hamming weights Secret sharing scheme

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New families of self-dual codes

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DESIGNS codeS AND CRYPTOGRAPHY 2021年第5期89卷 823-841页

作者： Sok, Lin Anhui Univ Sch Math Sci Hefei 230601 Anhui Peoples R China Royal Univ Phnom Penh Dept Math Phnom Penh 12156 Cambodia

Recently, the author has constructed families of MDS Euclidean self-dual codes from genus zero algebraic geometry (AG) codes. In the present correspondence, more families of optimal Euclidean self-dual codes from AG codes are explored. New families of MDS Euclidean self-dual codes of odd characteristic and those of almost MDS Euclidean self-dual codes are constructed explicitly from genus zero and genus one curves, respectively. More families of Euclidean self-dual codes are constructed from algebraic curves of higher genus.

关键词： Self-orthogonal code Self-dual code MDS code Almost MDS code Optimal code algebraic curve algebraic geometry code Differential algebraic geometry code

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Explicit Non-special Divisors of Small Degree, algebraic Geometric Hulls, and LCD codes from Kummer Extensions

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SIAM JOURNAL ON APPLIED ALGEBRA AND geometry 2024年第2期8卷 394-413页

作者： Moreno, Eduardo Camps Lopez, Hiram H. Matthews, Gretchen L. Virginia Tech Dept Math Blacksburg VA 24061 USA

In this paper, we consider the hull of an algebraic geometry code, meaning the intersection of the code and its dual. We demonstrate how codes whose hulls are algebraic geometry codes may be defined using only rational places of Kummer extensions (and Hermitian function fields in particular). Our primary tool is explicitly constructing non-special divisors of degrees g and g - 1 on certain families of function fields with many rational places, accomplished by appealing to Weierstrass semigroups. We provide explicit algebraic geometry codes with hulls of specified dimensions, producing along the way linear complementary dual (LCD) algebraic geometric codes from the Hermitian function field (among others) using only rational places and an answer to an open question posed by Ballet and Le Brigand for particular function fields. These results complement earlier work by Mesnager, Tang, and Qi that use lower-genus function fields as well as instances using places of a higher degree from Hermitian function fields to construct LCD codes and that of Carlet, Mesnager, Tang, Qi, and Pellikaan to provide explicit algebraic geometry codes with the LCD property rather than obtaining codes via monomial equivalences.

关键词： algebraic geometry code hull linear complementary dual maximal curve non-special divisor Weierstrass semigroup

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On Linear codes With One-Dimensional Euclidean Hull and Their Applications to EAQECCs

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IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第7期68卷 4329-4343页

作者： Sok, Lin Anhui Univ Sch Math Sci Hefei 230601 Anhui Peoples R China

The Euclidean hull of a linear code C is the intersection of C with its Euclidean dual C-perpendicular to. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented.

关键词： Hull MDS code almost MDS code optimal code algebraic curve algebraic geometry code entanglement-assisted quantum error-correcting code

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Quantum codes from one-point codes on norm-trace curves

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2022年第5期14卷 1179-1188页

作者： Kim, Boran Kyungpook Natl Univ Dept Math Educ Teachers Coll 80 Daehakro Daegu 41566 South Korea

In this paper, we present quantum codes via algebraic geometry codes on norm-trace curves. We provide a lower bound of minimum Hamming distance for q-ary quantum code, where q = 2(e) (e >= 3). In order to get this, we determine Feng-Rao function values for the elements of Weierstrass semigroups on norm-trace curves. We present the order-bound on the minimum Hamming distance of one-point dual codes. Furthermore, we give a certain increasing sequence of one-point codes on norm-trace curves. We construct quantum codes from the sequence of one-point codes via the CSS construction. These give a better lower bound on the minimum Hamming distance of q-ary quantum code than some previous results.

关键词： Norm-trace curve algebraic geometry code One-point code Dual code Quantum code

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