On deep holes of generalized projective Reed-Solomon codes

Abstract

Determining deep holes is an important topic in decoding Reed-Solomon codes. Let $l\ge 1$ be an integer and $a_1,\dots ,a_l$ be arbitrarily given $l$ distinct elements of the finite field ${\mathbf{F}}_q$ of $q$ elements with the odd prime number $p$ as its characteristic. Let $D={\mathbf{F}}_q\backslash \{a_1,\dots ,a_l\}$ and $k$ be an integer such that $2\le k\le q-l-1$. In this paper, we study the deep holes of generalized projective Reed-Solomon code ${\mathrm{GPRS}}_q(D,k)$ of length $q-l+1$ and dimension $k$ over ${\mathbf{F}}_q$. For any $f(x)\in {\mathbf{F}}_q[x]$, we let $f(D)=(f(y_1),\dots ,f(y_{q-l}))$ if $D=\{y_1,...,y_{q-l}\}$ and $c_{k-1}(f(x))$ be the coefficient of $x^{k-1}$ of $f(x)$. By using Dür's theorem on the relation between the covering radius and minimum distance of ${\mathrm{GPRS}}_q(D,k)$, we show that if $u(x)\in {\mathbf{F}}_q[x]$ with $°(u(x))=k$, then the received codeword $(u(D),c_{k-1}(u(x)))$ is a deep hole of ${\mathrm{GPRS}}_q(D,k)$ if and only if the sum $\sum _{y\in I}y$ is nonzero for any subset $I\subseteq D$ with $\#(I)=k$. We show also that if $j$ is an integer with $1\le j\le l$ and $u_j(x):={\lambda }_j(x-a_j{)}^{q-2}+{\nu }_j{x}^{k-1}+f_{\le k-2}^{(j)}(x)$ with ${\lambda }_j\in {\mathbf{F}}_q^{\ast }$, ${\nu }_j\in {\mathbf{F}}_q$ and $f_{\le k-2}^{(j)}(x)\in {\mathbf{F}}_q[x]$ being a polynomial of degree at most $k-2$, then $(u_j(D),c_{k-1}(u_j(x)))$ is a deep hole of ${\mathrm{GPRS}}_q(D,k)$ if and only if the sum $\left(\genfrac{}{}{0ex}{}{q-2}{k-1}\right)(-a_j{)}^{q-1-k}\prod _{y\in I}(a_j-y)+e$ is nonzero for any subset $I\subseteq D$ with $\#(I)=k$, where $e$ is the identity of the group ${\mathbf{F}}_q^{\ast }$. This implies that $(u_j(D),c_{k-1}(u_j(x)))$ is a deep hole of ${\mathrm{GPRS}}_q(D,k)$ if $p|k$.

16 pages