The Re-Encoding Transformation in Algebraic List-Decoding of Reed-Solomon Codes

TL;DR

This paper introduces a re-encoding and coordinate transformation technique that simplifies the bivariate polynomial interpolation step in algebraic list-decoding of Reed-Solomon codes, significantly reducing computational complexity.

Contribution

The paper presents a novel re-encoding transformation that reduces the size of the interpolation problem in algebraic list-decoding of Reed-Solomon codes, with a rigorous proof of equivalence.

Findings

Significant reduction in interpolation complexity

Efficient factorization on reduced problems

Theoretical proof of problem equivalence

Abstract

The main computational steps in algebraic soft-decoding, as well as Sudan-type list-decoding, of Reed-Solomon codes are bivariate polynomial interpolation and factorization. We introduce a computational technique, based upon re-encoding and coordinate transformation, that significantly reduces the complexity of the bivariate interpolation procedure. This re-encoding and coordinate transformation converts the original interpolation problem into another reduced interpolation problem, which is orders of magnitude smaller than the original one. A rigorous proof is presented to show that the two interpolation problems are indeed equivalent. An efficient factorization procedure that applies directly to the reduced interpolation problem is also given.