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}, \infty\} $, where $ \{x_1, \cdots, x_{n-1}\}\subsetneq \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_{\leq k-2}(x) $, where $ \lambda\in {\mathbb{F}}_q^* $, $ \delta\in{\mathbb{F}}_q\setminus \{x_1, \cdots, x_{n-1}\} $, $ \nu\in {\mathbb{F}}_q $ and $ f_{\leq{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 $ {\boldsymbol 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 $ {\boldsymbol u} $ is $ \lambda (x-\delta)^{q-2}+u_{n}x^{k-1}+u_{\leq{k-2}}(x), $ where $ \lambda \in {\mathbb{F}}_q^* $, $ \delta\in{\mathbb{F}}_q\setminus \{x_1, \cdots, x_{n-1}\} $, and $ u_{\leq{k-2}}(x) $ is a polynomial over $ {\mathbb{F}}_q $ whose degree does not exceed $ k-2 $, if $ \frac{q-1}{2}\leq k < n\leq q-2 $.