New Deep Holes of Generalized Reed-Solomon Codes

TL;DR

This paper introduces a concise method to identify new deep holes in generalized Reed-Solomon codes, extending previous results and providing insights into the structure of deep holes for specific code parameters.

Contribution

It presents a novel, simplified approach to find deep holes in generalized Reed-Solomon codes, including new classes for codes over fields with even characteristic and results on extended codes.

Findings

New class of deep holes for generalized Reed-Solomon codes

Deep holes characterized by polynomials of degree k+1 and k+2

Polynomials of degree k+2 do not define deep holes in extended codes

Abstract

Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong \cite{WH} found a new class of deep holes for standard Reed-Solomon codes. In the present paper, we give a concise method to obtain a new class of deep holes for generalized Reed-Solomon codes. In particular, for standard Reed-Solomon codes, we get the new class of deep holes given in \cite{WH}. Li and Wan \cite{L.W1} studied deep holes of generalized Reed-Solomon codes $GRS_{k}(\f,D)$ and characterized deep holes defined by polynomials of degree $k+1$. They showed that this problem is reduced to be a subset sum problem in finite fields. Using the method of Li and Wan, we obtain some new deep holes for special Reed-Solomon codes over finite fields with even characteristic. Furthermore, we study deep holes of the extended Reed-Solomon code, i.e., $D=\f$ and show polynomials of degree $k+2$ can not define deep holes.