Deep Holes of Twisted Reed-Solomon Codes
Weijun Fang, Jingke Xu · 2024
The deep holes of a linear code are the vectors achieving maximum error distance to the code. There has been a lot of work on the deep holes of Reed -Solomon codes. In this paper, we consider the deep holes of a class of twisted Reed -Solomon codes. The covering radius and a standard class of deep holes of twisted Reed-Solomon codes TRS$k(\mathcal{A},\ \eta)$are obtained for a general evaluation set$\mathcal{A}\subseteq \mathbb{F}_{q}$• Furthermore, when$q=2^{m}\geq 8$, we prove that there are no other deep holes of the full-length twisted Reed-Solomon codes TRS$k(\mathbb{F}_{q},\ \eta)$for$\displaystyle \frac{3}{4}q-1\leq k\leq{q}-4$, and we also completely determine their deep holes for$q-3\leq k\leq q-1$•