 R < C, it is a poor choice, since we must use n - k = 3 parity bits for k = 4 payload bits, which represents a heavy price for a bit-safe communication. But let us consider this example for the sake of illustration. As a Hamming code of distance dmin = 3, we have learnt that it has the capability of correcting no more than (dmin - 1)/2=1 bit errors out of a seven-bit block. If a single error occurs in a block sequence of 1001 bits (143 blocks), corresponding to BER 2 X 10-3, this error will be absolutely corrected, with a resulting BER of 0. If, in a block sequence of 2002 bits, exactly two errors occur within the same block, then only one error will be corrected, and the corrected BER will be reduced to BER 2 0.5 X 10-3 = = x x 10-4. Since the condition BER 2 0 is not achieved when there is more than one error, this code is, therefore, not optimal in view of the CCT, despite the fact that it satisfies the condition R < C. Note that the actual corrected error rate (which takes into account all error-event possibilities) is BER < X 10-6, which is significantly smaller yet not identical to zero, and is left as an exercise to show. Channel capacity and coding theorem Second, the CCT points to the existence of better codes (n, nR) with R < 0.9985 of which (255, 231), with R = 231/255 = 0.905 < C, for instance, is an eligible one. From Chapter 11, it is recognized as the Reed-Solomon code RS(255, 231) = (n = 2m - 1, k = n - 2t) with m = 8, t = 12, and code rate R = 231/255 = 0.905 < C. As we have seen, this RS code is capable of correcting up to t = 12 bit errors per 255- bit block, yielding, in this case, absolute error correction. Yet the finite possibility of getting 13 errors or more within a single block mathematically excludes the possibility of achieving BER = 0. The probability p(e > 12) of getting more than 12 bit errors in a single block is given by the formula (13.40) =p(13)+p(14)+...+p(255), = with (as assumed here) E = 10-3. Retaining only the first term p(13) of the expansion (13 errors exactly), we obt