Fast Decoding of Dual Multipoint Codes From Algebraic Curves Up to the Kirfel–Pellikaan Bound

Abstract

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 $d_{\mathrm{ Goppa}}$ . Although we can guarantee theoretically that every error vector of weight only up to $({1}/{2})(d_{\mathrm{ Goppa}}-g)$ 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 $({1}/{2})d_{\mathrm{ Goppa}}$ . 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 ${\mathcal{ O}}(a_{1}n^{2})$ , 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, ${\mathcal{ O}}(n^{({7}/{3})})$ for Hermitian codes. This complexity is less than the complexity ${\mathcal{ O}}(a_{1}gn^{2})$ of Lee’s method for decoding dual multipoint codes as a unique alternative.