) and (ii), respectively. The third equality, w(X+Z) = d(X, Z), comes from the definition of the Hamming distance, which represents the number of bit positions by which two blocks or vectors differ from each other; each of the nonzero bits in the vector X + Z represents such positions, which number as w(X+Z). Assume next that E1 is the error vector of the transmitted vector X. The received vector is, therefore: Y=X+E. (J10) The Hamming distances d(X, Y) and d(X, Z) are found using results in Eq. (J10) and property (i), respectively, as follows: =w(X+Y)=w(X-Y)=w(E1), (J11) d(Y,Z)=w(Y+Z)=w(X+E1+Z)= w(X- E1 Z) = w(E2). (J12) We assume next that the error pattern associated with E1 corresponds to a number of errors W (E1) = e strictly greater than t, i.e., e>t ore >1+ 1. Based on this assumption and after Eqs. (J3) and (J9), we have dmin=w(E1)+w(E2)<2t+2 w(E2)<t+1. (J13) t+1+w(E2) w(E1) + w(E2) < 2t +2 Substituting w(E1) z + + and w(E2) t + 1 in Eqs. (J11) and (J12), we obtain finally: d(Y, Z) < d(X, Y). (J14) d(Y, Z) < + + 1 The above result indicates that there exists an error pattern with e>t errors, which results in a transmitted vector Y closer to an incorrect vector Z than the transmitted vector X. As a consequence, based on the principle of maximum-likelihood decoding, the code will output an incorrectly decoded block. This provides the proof that the code is capable of correcting any error pattern with number of errors e <t, or e < {(dmin - 1)/2}, where the brackets indicate the integer contained in the argument. Appendix K (Chapter 13) Capacity of binary communication channels In this appendix, I provide the solution of the maximization problem for mutual infor- mation, which defines the channel capacity: C = max H(X;Y), as applicable to the general case of a binary communication channel (symmetric or asymmetric), whose transition matrix is defined according to where a, b are real numbers in the interval [0, 1]. To recall, the channel mutual infor- mation H(X; Y) is given by I(X;Y)=H(Y)-H(YX) Thus, we must f