New Deep Holes of Generalized Reed-Solomon Codes

Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong [11] found a new class of deep holes for standard Reed-Solomon codes. In the present paper, we give a concise method to obtain a new class of deep holes for generalized Reed-Solomon codes. In particular, for standard Reed-Solomon codes, we get the new class of deep holes given in [11]. Li and Wan [6] studied deep holes of generalized Reed-Solomon codes GRSk(Fq,D) and characterized deep holes defined by polynomials of degree k + 1. They showed that this problem is reduced to be a subset sum problem in finite fields. Using the method of Li and Wan, we obtain some new deep holes for special Reed-Solomon codes over finite fields with even characteristic. Furthermore, we study deep holes of the extended Reed-Solomon code, i.e., D = Fq and show polynomials of degree k + 2 can not define deep holes. keywords: coding theory, Reed-Solomon codes, list decoding, deep holes, multiplicative character, quadratic equation.