Deep Holes of Twisted Reed-Solomon Codes

The deep holes of a linear code are the vectors that achieve the maximum error distance to the code. There has been extensive research on the topic of deep holes in Reed-Solomon codes. 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. The covering radius and a standard class of deep holes of twisted Reed-Solomon codes TRSₖ(A, θ) are obtained for a general evaluation set A ⊆ Fq. Furthermore, we consider the problem of determining all deep holes of the full-length twisted Reed-Solomon codes TRSₖ(Fq, θ). Specifically, we prove that there are no other deep holes of TRSₖ(Fq, θ) for 3q-8/4 ≤ k≤ q-4 when q is even, and 3q+2√q-74 ≤ k≤ q-4 when q is odd. We also completely determine their deep holes for q-3 ≤ k ≤ q-1.