On deep holes of generalized Reed-Solomon codes

TL;DR

This paper extends the understanding of deep holes in generalized Reed-Solomon codes by identifying a new class of such deep holes, building on previous results for standard Reed-Solomon codes.

Contribution

It introduces a new class of deep holes for generalized Reed-Solomon codes, broadening the known characterization beyond standard codes.

Findings

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

Extended previous results from standard to generalized Reed-Solomon codes.

Abstract

Determining deep holes is an important topic in decoding Reed-Solomon codes. In a previous paper [8], we showed that the received word $u$ is a deep hole of the standard Reed-Solomon codes $[q-1, k]_q$ if its Lagrange interpolation polynomial is the sum of monomial of degree $q-2$ and a polynomial of degree at most $k-1$. In this paper, we extend this result by giving a new class of deep holes of the generalized Reed-Solomon codes.