Deep holes of a class of twisted Reed-Solomon codes

Abstract

The deep hole problem is a fundamental problem in coding theory, and it has many important applications in code constructions and cryptography. The deep hole problem of Reed-Solomon codes has gained a lot of attention. As a generalization of Reed-Solomon codes, we investigate the problem of deep holes of a class of twisted Reed-Solomon codes in this paper. Firstly, we provide the necessary and sufficient conditions for $\mathbf{a}=(a_0,a_1,\cdots ,a_{n-k-1})\in {\mathbb{F}}_q^{n-k}$ to be the syndrome of some deep hole of $TR{S}_k(\mathcal{A},l,\hat{I}\cdot )$. Next, we consider the problem of determining all deep holes of the twisted Reed-Solomon codes $TR{S}_k({\mathbb{F}}_q^{\ast },k-1,\hat{I}\cdot )$. Specifically, we prove that there are no other deep holes of $TR{S}_k({\mathbb{F}}_q^{\ast },k-1,\hat{I}\cdot )$ for $\frac{3q+2\sqrt{q}-8}{4}\le k\le q-5$ when q is even, and $\frac{3q+3\sqrt{q}-5}{4}\le k\le q-5$ when q is odd. We also completely determine their deep holes for $q-4\le k\le q-2$ when $q$ is even.