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

Abstract

We present a decision tree solution for the most complicated step in decoding binary BCH codes, namely the computation of an error location polynomial over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ) from the syndrome vector of received data. We run Lin's 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. This latter method can provide an extremely fast decoder that is compatible with realization in VLSI.