On the deep holes of a class of Cauchy codes

Suppose ${\mathbb{F}}_q$ is a finite field having an odd characteristic. Let $D=\{x_1,\cdots ,x_{n-1},\mathrm{\infty }\}$, where $\{x_1,\cdots ,x_{n-1}\}⊊{\mathbb{F}}_q$. Assume that $k$ is an integer with $2\le k<n$. We show that $u(D)$ is a deep hole of Cauchy code $C(D,k)$ if $u(x)=\lambda (x-\delta {)}^{q-2}+\nu x^{k-1}+f_{\le k-2}(x)$, where $\lambda \in {\mathbb{F}}_q^{\ast }$, $\delta \in {\mathbb{F}}_q\setminus \{x_1,\cdots ,x_{n-1}\}$, $\nu \in {\mathbb{F}}_q$ and $f_{\le k-2}(x)\in {\mathbb{F}}_q[x]$ of a degree that does not exceed $k-2$. This expands the result shown in our previous paper. In particular, we also show that the received word $\mathit{u}=(u_1,\cdots ,u_n)\in {\mathbb{F}}_q^n$ is a deep hole of $C(D,k)$ if and only if the Lagrange interpolation polynomial of the first $n-1$ components of $\mathit{u}$ is $\lambda (x-\delta {)}^{q-2}+u_n{x}^{k-1}+u_{\le k-2}(x),$ where $\lambda \in {\mathbb{F}}_q^{\ast }$, $\delta \in {\mathbb{F}}_q\setminus \{x_1,\cdots ,x_{n-1}\}$, and $u_{\le k-2}(x)$ is a polynomial over ${\mathbb{F}}_q$ whose degree does not exceed $k-2$, if $\frac{q-1}{2}\le k<n\le q-2$.