On error distance of Reed-Solomon codes

The complexity of decoding the standard Reed-Solomon code is a well known open prob-lem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum estimate, we show that the error distance can be determined precisely when the degree of the received word is small. As an application of our method, we give a significant improvement of the recent bound of Cheng-Murray on non-existence of deep holes (words with maximal error distance).