On Deep Holes of Projective Reed-Solomon Codes over Finite Fields with Even Characteristic

Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding. In this paper, we obtain two classes of deep holes of projective Reed-Solomon codes over finite fields with even characteristic. That is, let Fq be finite field with even characteristic, k∈{2,q-2}, and let u(x) be the Lagrange interpolation polynomial of the first q components of the received vector u∈Fqq+1. Suppose that the (q+1)-th component of u is 0, and u(x)=λxk+f≤k-2(x),λxq-2+f≤k-2(x), where λ∈Fq*, and f≤k-2(x) is a polynomial over Fq with degree no more than k-2. Then the received vector u is a deep hole of projective Reed-Solomon codes PRS(Fq,k). In fact, our result partially solved an open problem on deep holes of projective Reed-Solomon codes proposed by Wan in 2020