Closed solution of Berlekamp´s algorithm for fast decoding of BCH codes

A decision tree solution is presented for the most complicated step in decoding binary BCH codes, namely, the computation of an error location polynomial over GF(2^m) from the syndrome vector of received data. The author runs S. Lin´s (1970) iterative version of the Berlekamp-Massey algorithm symbolically, keeping the results at each level in the form of branches of a binary decision tree. A decoder can then be constructed that uses the derived formulas to evaluate a decision variable at each level. Complete traversal of the tree using the decision variables leads to the correct polynomial coefficients for the received vector. The decoder can be implemented in a very straightforward way with a simple processor or program that performs extension field arithmetic, or it can be realized entirely in hardware using lookup tables for multiplications, inverses, and exponents, and exclusive OR operations for addition