On the Threshold of Maximum-Distance Separable Codes

TL;DR

This paper investigates the decoding threshold of linear q-ary error-correcting codes, especially Reed-Solomon codes, from an information-theoretic perspective, establishing sharp thresholds and lower bounds relevant for cryptography.

Contribution

It introduces an information-theoretic framework for understanding decoding thresholds and provides explicit lower bounds for MDS codes like Reed-Solomon codes.

Findings

Threshold effect is very sharp when code's minimal distance is high.

Established lower bounds on decoding thresholds for Reed-Solomon codes.

Computed the threshold for a motivating toy example.

Abstract

Starting from a practical use of Reed-Solomon codes in a cryptographic scheme published in Indocrypt'09, this paper deals with the threshold of linear $q$-ary error-correcting codes. The security of this scheme is based on the intractability of polynomial reconstruction when there is too much noise in the vector. Our approach switches from this paradigm to an Information Theoretical point of view: is there a class of elements that are so far away from the code that the list size is always superpolynomial? Or, dually speaking, is Maximum-Likelihood decoding almost surely impossible? We relate this issue to the decoding threshold of a code, and show that when the minimal distance of the code is high enough, the threshold effect is very sharp. In a second part, we explicit lower-bounds on the threshold of Maximum-Distance Separable codes such as Reed-Solomon codes, and compute the threshold for the toy example that motivates this study.