Deep Holes of Twisted Reed-Solomon Codes

Abstract

The deep holes of a linear code are the vectors that achieve the maximum error distance (covering radius) to the code. {Determining the covering radius and deep holes of linear codes is a fundamental problem in coding theory. In this paper, we investigate the problem of deep holes of twisted Reed-Solomon codes.} The covering radius and a standard class of deep holes of twisted Reed-Solomon codes ${\rm TRS}_k(\mathcal{A}, \theta)$ are obtained for a general evaluation set $\mathcal{A} \subseteq \mathbb{F}_q$. Furthermore, we consider the problem of determining all deep holes of the full-length twisted Reed-Solomon codes ${\rm TRS}_k(\mathbb{F}_q, \theta)$. For even $q$, by utilizing the polynomial method and Gauss sums over finite fields, we prove that the standard deep holes are all the deep holes of ${\rm TRS}_k(\mathbb{F}_q, \theta)$ with $\frac{3q-4}{4} \leq k\leq q-4$. For odd $q$, we adopt a different method and employ the results on some equations over finite fields to show that there are also no other deep holes of ${\rm TRS}_k(\mathbb{F}_q, \theta)$ with $\frac{3q+3\sqrt{q}-7}{4} \leq k\leq q-4$. In addition, for the boundary cases of $k=q-3, q-2$ and $q-1$, we completely determine their deep holes using results on certain character sums.