On deep holes of generalized projective Reed-Solomon codes

Abstract. Determining deep holes is an important topic in decoding Reed-Solomon codes. Cheng and Murray, Li and Wan, Wu and Hong investigated the error distance of generalized Reed-Solomon codes. Recently, Zhang and Wan explored the deep holes of projective Reed-Solomon codes. Let l ≥ 1 be an integer and a1, . . . , al be arbitrarily given l distinct elements of the finite field Fq of q elements with the odd prime number p as its characteristic. Let D = Fq\{a1, . . . , al} and k be an integer such that 2 ≤ k ≤ q− l−1. In this paper, we study the deep holes of generalized projective Reed-Solomon code GPRSq(D, k) of length q− l+1 and dimension k over Fq. For any f(x) ∈ Fq[x], we let f(D) = (f(y1), . . . , f(yq−l)) if D = {y1, ..., yq−l} and ck−1(f(x)) be the coefficient of x k−1 of f(x). By using Dür’s theorem on the relation between the covering radius and minimum distance of GPRSq(D, k), we show that if u(x) ∈ Fq[x] with deg(u(x)) = k, then the received codeword (u(D), ck−1(u(x))) is a deep hole of GPRSq(D, k) if and only if the sum ∑