=1010111, Z2=0100001, Z3=0011110. 11.3 (M): With the Hamming code (n,k) = (7,4) described in the text, whose block codewords are listed in Table 11.1 and syndrome vectors are listed in Table 11.2, assume two bit errors in the received blocks. To how many possible error patterns do each of the syndrome vectors correspond? 11.4 (M): Calculate the probability that the Hamming code (n,k)= (7, 4) described in the text (with block codewords listed in Table 11.1) fails to detect any multiple 11.6 Exercises bit errors, while assuming single-bit-error probabilities of p= 10-2, 10-3, 10-4, and 10-5. 11.5 (T): Define a Hadamard code generated from a 4 x 4 matrix, then calculate the corresponding generator and parity-check matrices. 11.6 (M): Construct a cyclic code from the generator polynomial and tabulate the corresponding codewords. 11.7 (M): Construct the systematic nonrecursive convolutional code (n,k) = (3, 1) of constraint length m = 3 corresponding to the encoder Draw the corresponding coding tree for up to four input bits a, b, C, d. Channel entropy This relatively short chapter on channel entropy describes the entropy properties of communication channels, of which I have given a generic description in Chapter 11 concerning error-correction coding. It will also serve to pave the way towards probably the most important of all Shannon's theorems, which concerns channel coding, as described in the more extensive Chapter 13. Here, we shall consider the different basic communication channels, starting with the binary symmetric channel, and continuing with nonbinary, asymmetric channel types. In each case, we analyze the channel's entropy characteristics and mutual information, given a discrete source transmitting symbols and information thereof, through the channel. This will lead us to define the symbol error rate (SER), which corresponds to the probability that symbols will be wrongly received or mistaken upon reception and decoding. Binary symmetric channel The concept of the communication channel was introduced in C