Some results on deep holes of generalized projective Reed-Solomon codes

Abstract

Let $l\ge 1$ be an integer and $a_1,\ldots,a_l$ be arbitrarily given $l$ distinct elements of the finite field ${\bf F}_q$ of $q$ elements with the odd prime number $p$ as its characteristic. Let $D={\bf F}_q\backslash\{a_1,\ldots,a_l\}$ and $k$ be an integer such that $2\le k\le q-l-1$. For any $f(x)\in {\bf F}_q[x]$, we let $f(D)=(f(y_1),\ldots,f(y_{q-l}))$ if $D=\{y_1, ..., y_{q-l}\}$ and $c_{k-1}(f(x))$ be the coefficient of $x^{k-1}$ of $f(x)$. In this paper, by using Dür's theorem on the relation between the covering radius and minimum distance of the generalized projective Reed-Solomon code ${\rm GPRS}_q(D, k)$, we show that if $u(x)\in {\bf F}_q[x]$ with $\deg u(x)=k$, then the received word $(u(D), c_{k-1}(u(x)))$ is a deep hole of ${\rm GPRS}_q(D, k)$ if and only if $\sum\limits_{y\in I}y\ne 0$ for any subset $I\subseteq D$ with $\#(I)=k$. We show also that if $j$ is an integer with $1\leq j\leq l$ and $u_j(x):= \lambda_j(x-a_j)^{q-2}+\nu_j x^{k-1}+f_{\leq k-2}^{(j)}(x)$ with $\lambda_j\in {\bf F}_q^*$, $\nu_j\in {\bf F}_q$ and $f_{\leq{k-2}}^{(j)}(x)\in{\bf F}_q[x]$ being a polynomial of degree at most $k-2$, then $(u_j(D), c_{k-1}(u_j(x)))$ is a deep hole of ${\rm GPRS}_q(D, k)$ if and only if $\binom{q-2}{k-1}(-a_j)^{q-1-k}\prod\limits_{y\in I}(a_j-y)+e\ne 0$ for any subset $I\subseteq D$ with $\#(I)=k$, where $e$ is the identity of ${\bf F}_q^*$. Furthermore, $(u({\bf F}_q^*), c_{k-1}(u(x)))$ is not a deep hole of the primitive projective Reed-Solomon code ${\rm PPRS}_q({\bf F}_q^*, k)$ if $\deg u(x)=k$, and $(u({\bf F}_q^*),\delta)$ is a deep hole of ${\rm PPRS}_q({\bf F}_q^*, k)$ if $u(x)=\lambda x^{q-2}+\delta x^{k-1}+f_{\leq{k-2}}(x)$ with $\lambda\in {\bf F}_q^*$ and $\delta\in {\bf F}_q$.

Keywords

• generalized projective Reed-Solomon codes
• MDS codes
• deep holes
• Lagrange interpolation polynomial