Maximum-Likelihood Decoding of Reed-Solomon Codes is NP-hard

Title: Maximum-Likelihood Decoding of Reed-Solomon Codes is NP-hard

Research Question: Can maximum-likelihood decoding be made easier for any nontrivial family of codes with algebraic structure?

Methodology: The researchers used the technique of reduction, which is a common method in complexity theory to prove that a problem is NP-hard. They reduced the maximum-likelihood decoding problem of general linear codes to a problem known as Three-Dimensional Matching, which is an NP-complete problem. They then applied this reduction to Reed-Solomon codes, a specific type of code with nontrivial algebraic structure.

Results: The main result of this paper is that maximum-likelihood decoding of Reed-Solomon codes is NP-hard. This means that the problem of finding the most likely transmitted message given a received message and a code with certain properties is as hard as any other problem in NP (the class of problems that can be solved in polynomial time if given unlimited computational resources).

Implications: This result is significant because it shows that maximum-likelihood decoding remains hard even for a specific family of codes with nontrivial algebraic structure. This is in contrast to the general case of linear codes, for which maximum-likelihood decoding is known to be NP-hard. The result also contributes to the ongoing effort to understand the complexity of coding theory problems and to find efficient algorithms for decoding.

In practical terms, this result implies that decoding Reed-Solomon codes, which are widely used in error-correcting codes, cannot be done efficiently with current algorithms. This may lead to the development of new decoding algorithms or the exploration of other types of codes with better decoding properties.

Link to Article: https://arxiv.org/abs/0405005v1 Authors: arXiv ID: 0405005v1