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

Abstract

We consider a list decoding algorithm recently proposed by Pellikaan-Wu [8] for q-ary Reed-Muller codes R M q(l, m, n) of length n les q<sup>m</sup> when I les q. A simple and easily accessible correctness proof is given which shows that this algorithm achieves a relative error-correction radius of taules (1 - radiclq<sup>m-1</sup> /n). This is an improvement over the proof using one-point Algebraic-Geometric codes given in [8]. 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 &lt; Pi<sub>i=1</sub> <sup>m</sup>(1 - radick<sub>i</sub>/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.

DOI: 10.1109/ISIT.2007.4557130

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