List Decoding of Reed-Solomon Codes

This master thesis has been written for the completion of the author’s study of mathematics and deals with recent developments in coding theory. In particular, descriptions are given of recent algorithms for generating a list of all words form a Reed–Solomon code that are close to a received word. These algorithms may be adapted so that they can take into account the reliability information of distinct symbols. By simulation, the performance of various decoding algorithms for a specific code on the AWGN channel are compared. c © Koninklijke Philips Electronics N.V. 2003 iii PR-TN-2003-00628 Unrestricted Conclusions: • List decoding is a good way to improve on the performance of bounded distance decoding, especially for low rate codes. The algorithm of Sudan and Guruswami can be used to find a list of possibly sent codewords. This algorithm can also be extended so that reliability information can be taken into account. Koetter and Vardy have developed an efficient algorithm aimed at transforming reliability information to input for the algorithm of Guruswami and Sudan. It is proven that, given a bound on the desired complexity, the algorithm of Koetter and Vardy gives an optimal result for the number of errors that can be corrected. • A bivariate polynomial Q that has points (x j , yi) as zeroes of order m i j can be found by solving a system of C =i, j mi j (mi j +1)/2 linear equations. This takes O(C3) operations. Nielsen’s interpolation algorithm finds such a polynomial in O(bC2) operations, where b is an upper bound on the list size. This bound is always less than C. • The algorithm of Roth and Ruckenstein iteratively finds coefficients of polynomials f (X), such that Y − f (X) is a factor of a given polynomial Q(X,Y ). This method is more efficient than complete factorisation of a bivariate polynomial. • By simulation it is established that the Guruswami-Sudan algorithm gives good results for an [15, 3] Reed-Solomon code, especially if the algorithm of Koetter and Vardy is used to find interpolation points and their multiplicities from reliability information. In this case, if the number of interpolation points is limited to 15 all with multiplicity 1, the results are comparable with those of Generalised Minimum Distance decoding. However, the complexity is larger than that of GMD decoding. iv c © Koninklijke Philips Electronics N.V. 2003 Unrestricted PR-TN-2003-00628