2). Minimizing the SER is, therefore, a matter of using an input probability distribution such that p(x2) p(x1) < 1. In terms of coding, this means that the message sequences to be transmitted through the communications channel should contain the smallest possible Channel entropy number of symbols X2, or a number that should not exceed some critical threshold. For instance, given the constraint E = p(y1/x2) = 0.25 and a target error rate of SER = 10-2, we should impose for the code p(x2) < SER/E = 10-2/0.25 = 0.04. The corresponding code should not have more than 4% of X2 symbols in any input codeword sequence. Such a code has no reason to be optimal in terms of the efficient use of bits to transmit information. The point here is simply to illustrate that given a noisy communication channel, the SER can be made arbitrarily small through an adequate choice of input source coding. This conclusion anticipates Shannon's definition of channel capacity, which is described in Chapter 13. Through the above example, we have seen that the channel SER can be minimized by the adequate choice of source code. The issue of SER minimization should not be confused with the principle of error-correction codes (ECC), which were described in Chapter 11. The principle of ECC codes is to reduce the channel SER to an arbitrarily small or negligibly small level. As we have learnt, error detection and correction are performed by means of extra redundancy or parity bits, which are included in block codewords. Such bits represent overhead information, which is meant to detect and cor- rect errors automatically (up to some maximum), and which is removed once these tasks are accomplished. Error-correction codes, thus, make it possible to achieve transition matrices effectively approaching the identity matrix in Eq. (12.3) or Eq. (12.4) with E being made arbitrarily small. Exercises 12.1 (B): Show that any binary symmetric channel with X, Y as input and output sources has an invariant probability vector P(X) such that P(Y) = P(X). 12.2 (B): D