New List Decoding Algorithms for Reed-Solomon and BCH Codes

TL;DR

This paper introduces new list decoding algorithms for Reed-Solomon and BCH codes using rational curve fitting, achieving Johnson bound error correction with improved complexity over existing methods.

Contribution

It presents novel list decoding algorithms that match the Johnson bound and offer lower complexity compared to Guruswami-Sudan, with specific improvements for Reed-Solomon and BCH codes.

Findings

Corrects up to Johnson bound errors for Reed-Solomon codes

Achieves quadratic complexity with derivative list correction

Corrects up to Johnson bound errors for BCH codes

Abstract

In this paper we devise a rational curve fitting algorithm and apply it to the list decoding of Reed-Solomon and BCH codes. The proposed list decoding algorithms exhibit the following significant properties. 1 The algorithm corrects up to $n(1-\sqrt{1-D})$ errors for a (generalized) $(n, k, d=n-k+1)$ Reed-Solomon code, which matches the Johnson bound, where $D\eqdef \frac{d}{n}$ denotes the normalized minimum distance. In comparison with the Guruswami-Sudan algorithm, which exhibits the same list correction capability, the former requires multiplicity, which dictates the algorithmic complexity, $O(n(1-\sqrt{1-D}))$, whereas the latter requires multiplicity $O(n^2(1-D))$. With the up-to-date most efficient implementation, the former has complexity $O(n^{6}(1-\sqrt{1-D})^{7/2})$, whereas the latter has complexity $O(n^{10}(1-D)^4)$. 2. With the multiplicity set to one, the derivative list correction capability precisely sits in between the conventional hard-decision decoding and the optimal list decoding. Moreover, the number of candidate codewords is upper bounded by a constant for a fixed code rate and thus, the derivative algorithm exhibits quadratic complexity $O(n^2)$. 3. By utilizing the unique properties of the Berlekamp algorithm, the algorithm corrects up to $\frac{n}{2}(1-\sqrt{1-2D})$ errors for a narrow-sense $(n, k, d)$ binary BCH code, which matches the Johnson bound for binary codes. The algorithmic complexity is $O(n^{6}(1-\sqrt{1-2D})^7)$.