On Algebraic Decoding of $q$-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.

💡 Research Summary

This paper revisits the list‑decoding algorithm for $q$‑ary Reed‑Muller (RM) codes originally proposed by Pellikaan and Wu (PW2005) and provides a new, more transparent correctness proof. The authors focus on RM codes $\mathcal{RM}_q(\ell,m,n)$ with length $n\le q^m$ and degree $\ell\le q$. By directly analysing the rank properties of the evaluation matrix of multivariate polynomials over a $q$‑ary grid, they show that any error pattern affecting at most $\tau n$ symbols, where
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