List decoding of repeated codes

Assuming that we have a soft-decision list decoding algorithm of a linear code, a new hard-decision list decoding algorithm of its repeated code is proposed in this article. Although repeated codes are not used for encoding data, due to their parameters, we show that they have a good performance with this algorithm. We compare, by computer simulations, our algorithm for the repeated code of a Reed-Solomon code against a decoding algorithm of a Reed-Solomon code. Finally, we estimate the decoding capability of the algorithm for Reed-Solomon codes and show that performance is somewhat better than our estimates

Decoding of Repeated-Root Cyclic Codes up to New Bounds on Their Minimum Distance

The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root cyclic codes and present a syndrome-based burst error decoding algorithm with guaranteed decoding radius based on an associated folded cyclic code. Furthermore, we present a third technique for bounding the minimum Hamming distance based on the embedding of a given repeated-root cyclic code into a repeated-root cyclic product code. A second quadratic-time probabilistic burst error decoding procedure based on the third bound is outlined. Index Terms Bound on the minimum distance, burst error, efficient decoding, folded code, repeated-root cyclic code, repeated-root cyclic product cod

Prefactor Reduction of the Guruswami-Sudan Interpolation Step

The concept of prefactors is considered in order to decrease the complexity of the Guruswami-Sudan interpolation step for generalized Reed-Solomon codes. It is shown that the well-known re-encoding projection due to Koetter et al. leads to one type of such prefactors. The new type of Sierpinski prefactors is introduced. The latter are based on the fact that many binomial coefficients in the Hasse derivative associated with the Guruswami-Sudan interpolation step are zero modulo the base field characteristic. It is shown that both types of prefactors can be combined and how arbitrary prefactors can be used to derive a reduced Guruswami-Sudan interpolation step.Comment: 13 pages, 3 figure

On Algebraic Decoding of qq-ary Reed-Muller and Product-Reed-Solomon Codes

We consider a list decoding algorithm recently proposed by Pellikaan-Wu \cite{PW2005} for $q$-ary Reed-Muller codes $\mathcal{RM}_q(\ell, m, n)$ of length $n \leq q^m$ when $\ell \leq q$. A simple and easily accessible correctness proof is given which shows that this algorithm achieves a relative error-correction radius of $\tau \leq (1 - \sqrt{{\ell q^{m-1}}/{n}})$. This is an improvement over the proof using one-point Algebraic-Geometric codes given in \cite{PW2005}. The described algorithm can be adapted to decode Product-Reed-Solomon codes. We then propose a new low complexity recursive algebraic decoding algorithm for Reed-Muller and Product-Reed-Solomon codes. Our algorithm achieves a relative error correction radius of $\tau \leq \prod_{i=1}^m (1 - \sqrt{k_i/q})$. This technique is then proved to outperform the Pellikaan-Wu method in both complexity and error correction radius over a wide range of code rates.Comment: 5 pages, 5 figures, to be presented at 2007 IEEE International Symposium on Information Theory, Nice, France (ISIT 2007