On deep holes of standard Reed-Solomon codes

TL;DR

This paper disproves a longstanding conjecture by identifying a new class of deep holes in standard Reed-Solomon codes, significantly expanding the known set of such problematic received words.

Contribution

It demonstrates that the conjecture by Cheng and Murray is false by constructing a new class of deep holes for Reed-Solomon codes over prime power fields.

Findings

Identifies a new class of deep holes in Reed-Solomon codes.

Shows the existence of at least 2(q-1)q^k deep holes under certain conditions.

Disproves the previous conjecture about the uniqueness of trivial deep holes.

Abstract

Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the Reed-Solomon code. For the standard Reed-Solomon codes $[p-1, k]_p$ with $p$ a prime, Cheng and Murray conjectured in 2007 that there is no other deep holes except the trivial ones. In this paper, we show that this conjecture is not true. In fact, we find a new class of deep holes for standard Reed-Solomon codes $[q-1, k]_q$ with $q$ a prime power of $p$. Let $q \geq 4$ and $2 \leq k\leq q-2$. We show that the received word $u$ is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree $q-2$ and a polynomial of degree at most $k-1$. So there are at least $2(q-1)q^k$ deep holes if $k \leq q-3$.