On Deep Holes of Projective Reed-Solomon Codes over Finite Fields with Even Characteristic

Abstract

Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding. In this paper, we obtain two classes of deep holes of projective Reed-Solomon codes over finite fields with even characteristic. That is, let [see formula in PDF] be finite field with even characteristic, [see formula in PDF], and let [see formula in PDF] be the Lagrange interpolation polynomial of the first [see formula in PDF] components of the received vector [see formula in PDF]. Suppose that the [see formula in PDF]-th component of [see formula in PDF] is 0, and [see formula in PDF],[see formula in PDF] where [see formula in PDF], and [see formula in PDF] is a polynomial over [see formula in PDF] with degree no more than [see formula in PDF]. Then the received vector [see formula in PDF] is a deep hole of projective Reed-Solomon codes [see formula in PDF]. In fact, our result partially solved an open problem on deep holes of projective Reed-Solomon codes proposed by Wan in 2020.