A discrete memoryless channel C consists of two discrete memoryless channels D and E, which are connected serially as shown in the following figure.
The input alphabet of D is . Both of the output alphabet of D and the input alphabet of E are . The output alphabet of E is . Let random variables , and be respectively on , and . The channel transition matrix for D and the channel transition matrix for E are given as
Answer the following questions.
(1) Compute the channel capacity of D.
(2) Assume that follows the probability distribution given below. Compute the mutual information . You must show its derivation.
We consider only the AND operation and the XOR (exclusive or) operation for the elements in . We define a word as an element in , each of which is represented as a row vector. Consider the liner codes generated with a matrix of rows and columns as , where is a word and is a codeword. Let . Answer the following questions.
(1) Show that holds for all , where is the row vector obtained by element-wise XOR of two row vectors and .
(2) For the set , prove that
where is the Hamming distance and .
(3) For the case that is given below, find a matrix of rows and columns such that generates a systematic code. Moreover, by using the matrices and , compute the parity check matrix for .
