ere is no patterning effect within the possible bit sequences, or between codewords, namely that bit errors are uncorrelated. In a single transmission event, if m errors occur in a block of n bits, the corresponding error probability is, therefore, given by pm (1 - p)n-m The number of ways the m errors can be arranged in the block is C'n = n!/[m!(n - m)!]. The total error probability is then Cm pm (1 - p)n-m. Consider now the corrected BER. To evaluate the upper bound for the corrected BER, we must take into account all error possibilities, but include the fact that all error patterns with a maximum of w = {(dmin - 1)/2} have been effectively corrected. The corresponding BER is then given by the summation over all error-pattern possibilities Such a simplified demonstration makes no pretence of being accurate or unique. Error correction from m = W + 1 to m = n, representing the minimum and maximum numbers of incor- rect bits, respectively, i.e., (11.28) It is seen from Eq. (11.28) that the BER upper-bound in the right-hand side can be expressed as the finite sum BERcorr BERw+1 + BERw+2 (11.29) BERm=Cmpm(1-p)n-m = (11.30) In such a sum, each term corresponds to events of increasing numbers of bit errors, which have fast-decreasing likelihoods. It is, therefore, possible to approximate the BER bound by the first few terms in the sum. The accuracy of such approximation wholly depends on the block length, the value of w and the uncorrected bit-error probability, p. We shall consider a practical and realistic example, which also illustrates the powerful impact of ECC in BER correction. Assume the block code RS(n = 27 - 1 = 127, k = 119), which corresponds to the minimum Hamming distance dmin = (n - k)/2 = 4. We are, therefore, interested in evaluating the BER contributions BER5, BER6, etc., in Eq. (11.29), which correspond to over four bits of noncorrectable errors. We choose a realistic-case situation where the bit error probability is p = 10-4, From Eq. (11.30), we get, for the first three uncorrected BER contributio