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Every Convolutional code is a goppa code

IEEE TRANSACTIONS ON INFORMATION THEORY 2013年第10期59卷 6628-6641页

作者： Iglesias Curto, Jose Ignacio Castaneda, Angel Luis Munoz Munoz Porras, Jose Maria Serrano Sotelo, Gloria Univ Salamanca Dept Math E-37008 Salamanca Spain Univ Salamanca IUFFyM E-37008 Salamanca Spain Free Univ Berlin Inst Math D-14195 Berlin Germany

Convolutional goppa codes (CGC) were defined in Appl. Algebra Eng. Comm. Comput., vol. 15, pp. 51-61, 2004 and IEEE Trans. Inf. Theory, vol. 52, 340-344, 2006. In this paper, we prove that every convolutional code is a CGC defined over a smooth curve over F-q(z) and we give an explicit construction of convolutional codes as CGC over the projective line P-Fq(z)(1). We characterize which convolutional codes are defined by a complete linear system over curves of genus 0, 1, and over hyperelliptic curves. We apply these results to provide detailed constructions of some linear block codes as goppa codes.

关键词： Algebraic curve complete linear system convolutional code goppa code rational point

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goppa type sum-rank codes

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2025年第2期17卷 421-431页

作者： Chen, Yaqi Cheng, Zhiqiang Chen, Hao Jinan Univ Coll Cyber Secur Guangzhou 510632 Guangdong Peoples R China

We construct goppa type sum-rank codes over Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document} with the matrix size nxn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times n$$\end{document}, directly from goppa codes or extended goppa codes over Fqn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q<^>n}$$\end{document} in the Hamming metric. Lower bounds on dimensions and minimum sum-rank distances of goppa type sum-rank codes are proved. The goppa type sum-rank codes offer great flexibility in block length and dimension while maintaining strong error-correcting capability compared to the best known sum-rank codes. Furthermore, we construct numerous distance-optimal binary sum-rank codes with variable block length and minimum sum-rank distance four.

关键词： Sum-rank code goppa code Distance-optimal sum-rank code

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Quantum error correction with goppa codes from maximal curves: Design, simulation, and performance

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DISCRETE MATHEMATICS ALGORITHMS AND APPLICATIONS 2025年第0期

作者： Nourozi, Vahid New Mexico State Univ Klipsch Sch Elect & Comp Engn Las Cruces NM 88003 USA

This paper characterizes goppa codes of certain maximal curves over finite fields defined by equations of the form yn = xm + x. We investigate algebraic geometric and quantum stabilizer codes associated with these maximal curves and propose modifications to improve their parameters. The theoretical analysis is complemented by extensive simulation results, which validate the performance of these codes under various error rates. We provide concrete examples of the constructed codes, comparing them with known results to highlight their strengths and trade-offs. The simulation data, presented through detailed graphs and tables, offers insights into the practical behavior of these codes in noisy environments. Our findings demonstrate that while the constructed codes may not always achieve optimal minimum distances, they offer systematic construction methods and interesting parameter trade-offs that could be valuable in specific applications or for further theoretical study.

关键词： goppa code finite fields algebraic geometry codes quantum stabilizer codes maximal curve

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goppa codeS FROMARTIN-SCHREIER FUNCTION FIELDS

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Chinese Annals of Mathematics,Series B 1994年第3期15卷 311-320页

作者： HAN WENBAO(Depatmellt of Mathematics, Sichuan Universityt Chengdu 610064, Sichuan, China.) SICHUAN UNIV DEPT MATHCHENGDU 610064PEOPLES R CHINA

A class of goppa codes is constructed by using Artin-Schreier function fields, of which thenumber of prime divisors of degree olle is obtained for some cases, and their minimum distance,duallty and selfeduality are discussed. At laSt the sublield subcode of Artin-Schreier code isinvestigated, the true dimension under certain conditions is given and the covering radius andminimum distance are estimated.

关键词： Artin-Schreier functioll field goppa code Self-duality.

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On Rational Interpolation-Based List-Decoding and List-Decoding Binary goppa codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2013年第6期59卷 3269-3281页

作者： Beelen, Peter Hoholdt, Tom Nielsen, Johan S. R. Wu, Yingquan Tech Univ Denmark Dept Appl Math & Comp Sci DK-2800 Lyngby Denmark Sandforce Inc Milpitas CA 95035 USA

We derive the Wu list-decoding algorithm for generalized Reed-Solomon (GRS) codes by using Grobner bases over modules and the Euclidean algorithm as the initial algorithm instead of the Berlekamp-Massey algorithm. We present a novel method for constructing the interpolation polynomial fast. We give a new application of the Wu list decoder by decoding irreducible binary goppa codes up to the binary Johnson radius. Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami-Sudan algorithm, immediately leading to equality in the decoding range and a duality in the choice of parameters needed for decoding, both in the case of GRS codes and in the case of goppa codes.

关键词： goppa code Johnson radius list decoding list size rational interpolation Reed-Solomon code

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An Efficient Decoding of goppa codes for the McEliece Cryptosystem

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FUNDAMENTA INFORMATICAE 2014年第4期133卷 387-397页

作者： Lim, Seongan Lee, Hyang-Sook Choi, Mijin Ewha Womans Univ Inst Math Sci Seoul 120750 South Korea Ewha Womans Univ Dept Math Seoul 120750 South Korea

The McEliece cryptosystem is defined using a goppa code, and decoding the goppa code is a crucial step of its decryption. Patterson's decoding algorithm is the best known algorithm for decoding goppa codes. Currently, the most efficient implementation of Patterson's algorithm uses a precomputation. In this paper, we modify Patterson's decoding algorithm so that one can remove the precomputation part while sustaining the best efficiency. Precomputations yield additional storage requirement to store the precomputed value which increases as the security level increases in McEliece cryptosystem. In the original decoding algorithm of Patterson, computing square root in a quotient field of polynomial ring over a finite field is necessary. In our modification, the computations are involved only in the arithmetics of polynomial ring over a finite field, not in the quotient field. This achieves better efficiency because one can remove polynomial reductions in the computations of quotient field.

关键词： McEliece Cryptosystem goppa code Patterson's algorithm square roots

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Minimum cyclotomic coset representatives and their applications to BCH codes and goppa codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2000年第7期46卷 2625-2628页

作者： Yue, DW Feng, GZ Nanjing Univ Posts & Telecommun Dept Telecommun Engn Nanjing 210003 Peoples R China Univ Manitoba Dept Elect & Comp Engn Winnipeg MB R3T 5V6 Canada

In this correspondence, we consider the minimum cyclotomic coset representatives and derive some of their properties. The results allow more precise estimates of the dimension of Bose-Chaudhuri-Hocquenghem (BCH) and classical goppa codes of a given designed minimum distance, and more precise estimates of the true designed distance of BCH codes and the minimum distance of classical goppa codes.

关键词： Bose-Chaudhuri-Hocquenghem (BCH) code cyclotomic coset designed distance dimension goppa code minimum cyclotomic coset representative minimum distance

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Binary quasi-cyclic goppa codes

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DESIGNS codeS AND CRYPTOGRAPHY 2000年第2期20卷 107-124页

作者： Bommier, G Blanchet, F Inst Natl Rech Informat & Automat Codes F-78153 Le Chesnay France

We obtain here a necessary and sufficient condition for a certain class of binary goppa code to be quasi-cyclic. We also give another sufficient condition which is easier to check. We define a class of quasi-cyclic goppa codes. We find the true dimension for a part of those quasi-cyclic codes. and also a class of extended quasi-cyclic codes the minimum distance of which is equal to the designed distance.

关键词： goppa code quasi-cyclic code linear code

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Patterson Algorithm for Decoding Separable Binary goppa codes

Patterson Algorithm for Decoding Separable Binary Goppa Code...

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Wave Electronics and its Application in Information and Telecommunication Systems (WECONF)

作者： Bezzateev, S., V Noskov, I. K. St Petersburg State Univ Aerosp Instrumentat St Petersburg Russia ITMO Univ St Petersburg Russia

In this paper, we consider the goppa codes, their decoding algorithms, their application in practice, and also the special properties of using the Patterson algorithm in the case of its use for decoding separable bina... 详细信息

ISBN: (纸本)9781728122885

关键词： goppa code Patterson algorithm separable polynomial

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Implementation of McEliece Based on Quasi-dyadic goppa codes for Embedded Devices

Implementation of McEliece Based on Quasi-dyadic Goppa Codes...

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4th International Workshop on Post-Quantum Cryptography

作者： Heyse, Stefan Ruhr Univ Bochum Horst Gortz Inst IT Secur D-44780 Bochum Germany

ISBN: (纸本)9783642254048

Most public-key cryptosystems frequently implemented have been proven secure on the basis of the presumed hardness of two mathematical problems: factoring the product of two large primes (FP) and computing discrete logarithms (DLP). At present, both problems are believed to be computationally infeasible with an ordinary computer. However, a quantum-computer having the ability to perform computations on a few thousand qbits could solve both problems using Shor's algorithm [23]. Although a quantum computer of this dimension has not been reported, development and cryptanalysis of alternative public-key cryptosystems seem suitable. To achieve acceptance and attention in practice, they have to be implemented efficiently. Furthermore, the implementations have to perform fast while keeping memory requirements low for security levels comparable to conventional schemes. The McEliece encryption and decryption do not require computationally expensive multiple precision arithmetic. Hence, it is predestined for an implementation on embedded devices. The major disadvantage of the McEliece public-key cryptosystem(PKC) is its very large public key of several hundred thousands bits. For this reason, the McEliece PKC has achieved little attention in the practice. Another disadvantage of the McEliece scheme, like many other schemes, is that it is not semantically secure. The quasi-dyadic McEliece variant proposed by Barreto and Misoczki addresses both problems. In this work we provide an implementation of this alternative public-key cryptosystem, which is semantically secure and uses a 40 times smaller public key and a five times smaller secret key compared to a previously published implementation [6].

关键词： McEliece goppa code Quasi-Dyadic Embedded Device Post-Quantum

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