On deep holes of generalized projective Reed-Solomon codes

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 $D={\bf F}q\backslash{a_1,\ldots,a_l}$and$k$beanintegersuchthat$2\le k\le q-l-1$. In this paper, we study the deep holes of generalized projective Reed-Solomon code ${\mathrm{G}\mathrm{P}\mathrm{R}\mathrm{S}}_q(D,k)$ 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 $D=\{y_1,\mathrm{.}\mathrm{.}\mathrm{.},y_{q-l}\}$ and $c_{k-1}(f(x))$ be the coefficient of $x^{k-1}$ of $f(x)$. By using D"ur’s theorem on the relation between the covering radius and minimum distance of ${\mathrm{G}\mathrm{P}\mathrm{R}\mathrm{S}}_q(D,k)$, we show that if $u(x)\in {\mathbf{F}}_q[x]$ with $\deg (u(x))=k$,thenthereceivedcodeword$(u(D), c{k-1}(u(x)))$isadeepholeof${\rm GPRS}q(D, k)$ifandonlyifthesum$\sum\limits{y\in I}y$ is nonzero for any subset $I\subseteq D$ with $\mathrm{\#}(I)=k$. We show also that if $j$ is an integer with $1\le j\le l$ and $u_j(x):= \lambda_j(x-a_j)^{q-2}+\nu_j x^{k-1}+f_{\leq k-2}^{(j)}(x)$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 ${\rm GPRS}q(D, k)$ifandonlyifthesum$\binom{q-2}{k-1}(-a_j)^{q-1-k}\prod\limits{y\in I}(a_j-y)+e$ is nonzero for any subset $I\subseteq D$ with $\mathrm{\#}(I)=k$, 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 ${\mathrm{G}\mathrm{P}\mathrm{R}\mathrm{S}}_q(D,k)$ if $p\mathrm{\mid }k$.