Deep holes of generalized Reed-Solomon codes
Jun Zhang, Qunying Liao, Fang‐Wei Fu · Scientia Sinica Mathematica · 2013
Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong found a new class of deep holes for standard Reed-Solomon codes. In the present paper, we give a simple method to obtain a new class of deep holes for generalized Reed-Solomon codes. In particular, for standard Reed-Solomon codes, we got the new class of deep holes. Li and Wan 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 characterize the deep holes defined by polynomials of degree k+2. And we show that polynomials of degree k+2 can not define deep holes.