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one-point codes using places of higher degree

IEEE TRANSACTIONS ON INFORMATION THEORY 2005年第4期51卷 1590-1593页

作者： Matthews, GL Michel, TW Clemson Univ Dept Math Sci Clemson SC 29634 USA

In IEEE Transactions on Information Theory, vol. 48, no. 2, pp. 535-537, Feb. 2002, Xing and Chen show that there exist algebraic-geometry (AG) codes from the Hermitian function field over F-q2 constructed using F-q2 -rational divisors which are improvements over the much-studied one-point Hermitian codes. In this correspondence, we construct such codes by using a place P of degree r > 1. This motivates a study of gap numbers and pole numbers at places of higher degree. In fact, the code parameters are estimated using the Weierstrass gap set of the place P and relating it to the gap set of the r-tuple of places of degree one lying over P in a constant field extension of degree r.

关键词： algebraic-geometry (AG) code degree of a place Hermitian function field one-point code Weierstrass gap set

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Algebraic-geometry codes, one-point codes, and evaluation codes

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DESIGNS codeS AND CRYPTOGRAPHY 2007年第2-3期43卷 137-145页

作者： Bras-Amoros, Maria Univ Autonoma Barcelona E-08193 Barcelona Spain

one-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed-Muller codes. Given an F-q-algebra A, an order function rho on A and given a surjective F-q-morphism of algebras phi: A -> F-q(n), the ith evaluation code with respect to A, rho, phi is defined as the code C-i = phi(f is an element of A : rho(f) <= i}). In this work it is shown that under a certain hypothesis on the F-q-algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over F-q and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G(i) with associated divisors Gamma(i) is the sequence of evaluation codes associated to some F-q-algebra A, some order function rho and some surjective morphism phi with {f is an element of A : rho(f) <= i} = L(Gamma(i)) if and only if it is a sequence of one-point codes.

关键词： algebraic-geometry code one-point code evaluation 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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A sequence of one-point codes from a tower of function fields

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DESIGNS codeS AND CRYPTOGRAPHY 2006年第3期41卷 251-267页

作者： Hasegawa, Takehiro Kondo, Shoichi Kurusu, Hidekazu Waseda Univ Sch Educ Dept Math Tokyo 169805 Japan Waseda Univ Grad Sch Sci & Engn Tokyo 1698555 Japan

We construct a sequence of one-point codes from a tower of function fields whose relative minimum distances have a positive limit. Our tower is characterized by principal divisors. We determine completely the minimum distance of the codes from the first field of our tower. These results extend those of Stichtenoth [IEEE Trans Inform Theory (1988), 34(15):1345-1348], Yang and Kumar [Lecture Notes in Mathematics, 1518, (1991), Springer-Verlag, Berlin Hidelberg New York, pp. 99-107], and Garcia [Comm. Algebra, 20(12): 3683-3689]. As an application, we show that the minimum distance corresponds to the Feng-Rao bound.

关键词： tower of function fields one-point code minimum distance Feng-Rao bound

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On the geometry of Hermitian one-point codes

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JOURNAL OF ALGEBRA 2014年 397卷 499-514页

作者： Ballico, Edoardo Ravagnani, Alberto Univ Trento Dept Math I-38123 Povo TN Italy Univ Neuchatel Inst Math CH-2000 Neuchatel Switzerland

Here we describe the algebraic geometry of one-point codes from the Hermitian curve. In particular, we employ zero-dimensional schemes in the plane to characterize the minimum-weight code-words of their dual codes, providing explicit formulas for their number. We discuss also some natural improvements of the duals of Hermitian one-point codes by means of geometric arguments. Finally, some cohomological tools are developed to characterize the small-weight codewords of such codes. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Hermitian curve Goppa code one-point code Minimum-weight codeword

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Fast Decoding of Multipoint codes from Algebraic Curves

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IEEE TRANSACTIONS ON INFORMATION THEORY 2014年第4期60卷 2054-2064页

作者： Sakata, Shojiro Fujisawa, Masaya Univ Electrocommun Tokyo 1920363 Japan Tokyo Univ Sci Tokyo 1628601 Japan

Multipoint codes are a broad class of algebraic geometry codes derived from algebraic functions, which have multiple poles and/or zeros on an algebraic curve. Thus, they are more general than one-point codes, which are an important class of algebraic geometry codes in the sense that they can be decoded efficiently using the Berlekamp-Massey-Sakata algorithm. We present a fast method for decoding multipoint codes from a plane curve, particularly a Hermitian curve. Our method with some adaptation can be applied to decode multipoint codes from a general algebraic curve embedded in the N-dimensional affine space F-q(N) over a finite field F-q, so that those algebraic geometry codes can be decoded efficiently if the dimension N of the affine space, including the defining curve is small.

关键词： Algebraic geometry codes algebraic curve multipoint code one-point code fast decoding vectorial BMS algorithm

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Parameter choices and a better bound on the list size in the Guruswami-Sudan algorithm for algebraic geometry codes

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DESIGNS codeS AND CRYPTOGRAPHY 2010年第2期54卷 181-187页

作者： Drake, Nathan Matthews, Gretchen L. Clemson Univ Dept Math Sci Clemson SC 29634 USA

Given an algebraic geometry code C-L(D, alpha P), the Guruswami-Sudan algorithm produces a list of all codewords in C-L(D, alpha P) within a specified distance of a received word. The initialization step in the algorithm involves parameter choices that bound the degree of the interpolating polynomial and hence the length of the list of codewords generated. In this paper, we use simple properties of discriminants of polynomials over finite fields to provide improved parameter choices for the Guruswami-Sudan list decoding algorithm for algebraic geometry codes. As a consequence, we obtain a better bound on the list size as well as a lower degree interpolating polynomial.

关键词： AG code List decoding one-point code Polynomial reconstruction

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Fast Decoding of Dual Multipoint codes From Algebraic Curves Up to the Kirfel-Pellikaan Bound

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IEEE TRANSACTIONS ON INFORMATION THEORY 2018年第6期64卷 4452-4466页

作者： Sakata, Shojiro Fujisawa, Masaya Univ Electrocommun Chofu Tokyo 1828585 Japan Tokyo Univ Sci Tokyo 1258585 Japan

The multipoint codes from algebraic curves are a broad class of algebraic geometry codes derived from algebraic functions, which have multiple poles/zeros on their defining curves. Each of them is defined as either a primal code or a dual code. The dual one-point codes which are viewed as a subclass can be decoded efficiently up to the Feng-Rao bound by using the Berlekamp-Massey-Sakata (BMS) algorithm with majority logic. Since a primal code is equivalent to a dual code, one can decode as either of them, while their decoding methods are different. Recently, we published a fast method for decoding primal multipoint codes from curves based on the vectorial BMS algorithm. But, that is neither for dual codes nor up to the Goppa bound d(Goppa). Although we can guarantee theoretically that every error vector of weight only up to (1/2)(d(Goppa)-g) can be corrected, where the integer g is the genus of the defining curve, the simulation shows that the method can correct most error patterns of weight up to (1/2) d(Goppa). In this paper we present a fast method for decoding dual multipoint codes from algebraic curves up to the Kirfel-Pellikaan bound, based on the vectorial BMS algorithm with majority logic, and show that algebraic geometry codes from generic algebraic curves can be decoded up to the Goppa bound efficiently. Similar to the case of one-point codes, the computational complexity of decoding is O(a(1)n(2)), where the integer a(1) is the minimum nonzero pole order of algebraic functions on the defining curve and the integer n is the code length, and in particular, O(n((7/3))) for Hermitian codes. This complexity is less than the complexity O(a(1)gn(2)) of Lee's method for decoding dual multipoint codes as a unique alternative.

关键词： code from curve multipoint code one-point code primal code dual code Goppa bound Kirfel-Pellikaan bound order bound fast decoding vectorial BMS algorithm majority voting

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Deformations and smoothability of certain AS monomial curves

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BEITRAGE ZUR ALGEBRA UND GEOMETRIE-CONTRIBUTIONS TO ALGEBRA AND GEOMETRY 2014年第2期55卷 557-575页

作者： oneto, Anna Tamone, Grazia Univ Genoa Dima Via Dodecaneso 35 I-16146 Genoa Italy

In this paper, we consider semigroups of embedding dimension five generated by arithmetic sequences. We prove these semigroups are Weierstrass by showing that the associated monomial curves are smoothable.

关键词： Numerical semigroup Arithmetic sequence Monomial curve Deformation Weierstrass semigroup one-point code

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Fast Decoding of Dual Multipoint codes From Algebraic Curves Up to the Kirfel-Pellikaan Bound

Fast Decoding of Dual Multipoint Codes From Algebraic Curves...

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9th International Workshop on Coding and Cryptography (WCC)

作者： Sakata, Shojiro Fujisawa, Masaya Univ Electrocommun Chofu Tokyo 1828585 Japan Tokyo Univ Sci Tokyo 1258585 Japan

关键词： code from curve multipoint code one-point code primal code dual code Goppa bound Kirfel-Pellikaan bound order bound fast decoding vectorial BMS algorithm majority voting

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