Maximum-likelihood decoding of Reed-Solomon Codes is NP-hard

TL;DR

This paper proves that maximum-likelihood decoding of Reed-Solomon codes is NP-hard, establishing computational difficulty even with unlimited preprocessing, which impacts decoding strategies in coding theory.

Contribution

It demonstrates that maximum-likelihood decoding remains NP-hard for Reed-Solomon codes, a significant algebraic family, extending known hardness results.

Findings

Maximum-likelihood decoding of Reed-Solomon codes is NP-hard.

Hardness persists even with unlimited preprocessing.

Strengthens previous results by Bruck and Naor.

Abstract

Maximum-likelihood decoding is one of the central algorithmic problems in coding theory. It has been known for over 25 years that maximum-likelihood decoding of general linear codes is NP-hard. Nevertheless, it was so far unknown whether maximum- likelihood decoding remains hard for any specific family of codes with nontrivial algebraic structure. In this paper, we prove that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes. We moreover show that maximum-likelihood decoding of Reed-Solomon codes remains hard even with unlimited preprocessing, thereby strengthening a result of Bruck and Naor.