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 ${\mathrm{TRS}}_k(A,\theta )$ are obtained for a general evaluation set $A\subseteq F_q$. Furthermore, we consider the problem of determining all deep holes of the full-length twisted Reed-Solomon codes ${\mathrm{TRS}}_k(F_q,\theta )$. 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 ${\mathrm{TRS}}_k(F_q,\theta )$ with $\frac{3q-4}{4}\le k\le q-4$. 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 ${\mathrm{TRS}}_k(F_q,\theta )$ with $\frac{3q+3\sqrt{q}-7}{4}\le k\le q-4$. 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.