Performance bound for parity-check coded partial-response channels

We consider maximum-likelihood decoder performance in additive white Gaussian noise for the serial concatenation of an outer code comprising multiple, independent odd-parity-check codes and an inner precoded dicode partial-response channel through a random interleaver. Using a technique proposed in Oberg and Siegel (1998, 2001) we derive an approximation of the average weight enumerator, assuming that code bit values within error events are independent, identically distributed, and equiprobable. We show that the union bound on word-error-rate based upon the approximate weight enumerator is actually an upper bound to that based upon the true average weight enumerator, for suitably large signal-to-noise ratios