BCH cyclic codes

BCH codes are a subclass of cyclic codes with many interesting properties. In many cases BCH codes are the best linear codes. For example, among all binary cyclic codes of odd length n with n≤125 the best cyclic code is always a BCH code except for two special cases. Reed-Solomon codes are also BCH codes and are widely used in communication devices and consumer electronics. Binary BCH codes were discovered by Hocquenghem and independently by Bose and Ray-Chaudhuri. They were extended to BCH codes over finite fields by Gorenstein and Zierler in 1961. In fact, BCH codes were introduced by Bush in 1952 in the context of orthogonal arrays. In the past sixty years, a lot of progress on the study of BCH codes has been made. The objective of this survey is to present the state-of-the-art of BCH cyclic codes and their applications.