Efficient Factorisation Algorithm for List Decoding Algebraic-Geometric and Reed-Solomon Codes

Abstract

The list decoding algorithm can outperform the conventional unique decoding algorithm by producing a list of candidate decoded messages. An efficient list decoding algorithm for algebraic-geometric (AG) codes and Reed-Solomon (RS) codes has been developed by Guruswami and Sudan, called the Guruswami-Sudan (GS) algorithm. The algorithm includes two steps: Interpolation and Factorisation. To implement interpolation, Koetter proposed an iterative polynomial construction algorithm for RS codes. By redefining a polynomial over algebraic function fields, Koetter's algorithm can also be applied to AG codes. To implement factorisation, Roth and Ruckenstein proposed an efficient algorithm for RS codes and later Wu and Siegel extended it to AG codes. Following on from their previous work, we propose a more general factorisation algorithm which can be applied to both AG and RS codes. This algorithm avoids rational function quotient calculations required by Wu and SiegePs algorithm, making it more efficient to implement. As well as employing this algorithm to list decode AG and RS codes this paper also presents the first simulation results evaluating the list decoding performance comparison between AG and RS codes of a similar code rate defined over the same finite field.