On error distance of received words with fixed degrees to Reed-Solomon code

TL;DR

This paper investigates the computational difficulty of decoding Reed-Solomon codes by analyzing the error distance of received words with fixed degrees, providing exact calculations for certain cases and implications for the deep hole problem.

Contribution

It introduces algebraic constructions to exactly determine error distances for specific degrees, advancing understanding of Reed-Solomon decoding complexity.

Findings

Exact error distance for degree k+1 received words to RS codes

Exact error distance for degree k+2 received words to RS codes

Implications for the deep hole problem in Reed-Solomon codes

Abstract

Under polynomial time reduction, the maximum likelihood decoding of a linear code is equivalent to computing the error distance of a received word. It is known that the decoding complexity of standard Reed-Solomon codes at certain radius is at least as hard as the discrete logarithm problem over certain large finite fields. This implies that computing the error distance is hard for standard Reed-Solomon codes. Using some elegant algebraic constructions, we are able to determine the error distance of received words whose degree is k+1 to the Standard Reed-Solomon code or Primitive Reed-Solomon code exactly. Moreover, we can precisely determine the error distance of received words of degree k+2 to the Standard Reed-Solomon codes. As a corollary, we can simply get the results of Zhang-Fu-Liao and Wu-Hong on the deep hole problem of Reed-Solomon codes.