On Hard and Soft Decision Decoding of BCH Codes

Cyclic codes have the advantage that it is only necessary to store one polynomial. The binary primitive BCH codes are cyclic and are created by choosing a subset of the cyclotomic cosets which can be done in various ways. We compare different BCH codes of the same coderate with different weight distributions, thus, are not equivalent by using different choices of cyclotomic cosets. We recall an old result from the sixties that any Reed-Muller code is equivalent to a particular BCH code extended by a parity bit. The motivation for decoding BCH codes is that they have possibly better parameters than Reed-Muller codes which are related in recent publications to polar codes. We present several hard and soft decision decoding schemes based on minimal weight codewords of the dual code, including information set decoding in case of a channel without reliability information. Different BCH codes of the same rate are compared and show different decoding performance and complexity. Some examples of hard decision decoding of BCH codes have the same decoding performance as maximum likelihood decoding. All presented decoding methods can be extended to include reliability information of a Gaussian channel for soft decision decoding. We show various simulation results for soft decision list information set decoding and analyze the influence of different aspects on the performance.