Bounds on Error-Correction Coding Performance

The theoretical performance of binary codes for the additive white Gaussian noise (AWGN) channel is discussed. In particular, Gallager’s coding theorem for binary codes is treated extensively. By assuming a binomial weight distribution for linear codes, it is shown that the decoder error probability performance of some of the best, known linear, binary codes is the same as the average performance of the ensemble of all randomly chosen, binary nonlinear codes having the same length and dimension. Assuming a binomial weight distribution, an upper bound is determined for the erasure performance of any code, and it is shown that this can be translated into an upper bound for code performance in the AWGN channel. Finally, theoretical bounds on the construction of error-correction codes are discussed.