Mathematics

CLC number: O 236.2

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

School of Big Data and Statistics, Sichuan Tourism University, Chengdu 610100, Sichuan, China

Received: 14 July 2022

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 ${\mathbb{F}}_q$ be finite field with even characteristic, $k\in \{\mathrm{2},q-\mathrm{2}\}$, and let $u(x)$ be the Lagrange interpolation polynomial of the first $q$ components of the received vector $\mathit{u}\in {\mathbb{F}}_q^{q+\mathrm{1}}$. Suppose that the $(q+\mathrm{1})$-th component of $\mathit{u}$ is 0, and $u(x)=\lambda x^k+f_{\le k-\mathrm{2}}(x)$,$\lambda x^{q-\mathrm{2}}+f_{\le k-\mathrm{2}}(x),$ where $\lambda \in {\mathbb{F}}_q^{\mathrm{*}}$, and $f_{\le k-\mathrm{2}}(x)$ is a polynomial over ${\mathbb{F}}_q$ with degree no more than $k-\mathrm{2}$. Then the received vector $\mathit{u}$ is a deep hole of projective Reed-Solomon codes $\mathrm{P}\mathrm{R}\mathrm{S}({\mathbb{F}}_q,k)$. In fact, our result partially solved an open problem on deep holes of projective Reed-Solomon codes proposed by Wan in 2020.