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Fast Decoding of multipoint codes from Algebraic Curves

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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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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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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Minimum Distance Decoding of General Algebraic Geometry codes via Lists

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IEEE TRANSACTIONS ON INFORMATION THEORY 2010年第9期56卷 4335-4340页

作者： Drake, Nathan Matthews, Gretchen L. N Greenville Univ Dept Math Tigerville SC 29688 USA Clemson Univ Dept Math Sci Clemson SC 29634 USA

Algebraic geometry codes are defined by divisors D and G on a curve over a finite field F. Often, G is supported by a single F-rational point and the resulting code is called a one-point code. Recently, there has been interest in allowing the divisor G to be more general as this can result in superior codes. In particular, one may obtain a code with better parameters by allowing G to be supported by m distinct F-rational points, where m > 1. In this paper, we demonstrate that a multipoint algebraic geometry code C may be embedded in a one-point code C'. Exploiting this fact, we obtain a minimum distance decoding algorithm for the multipoint code C. This is accomplished via list decoding in the one-point code C'.

关键词： Algebraic geometry (AG) code list decoding minimum distance decoding multipoint code

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