d as (01): f(u)=-ulogu-(1-u)log(1-u). 13.2 (M): Determine the bit capacity of a four-input, four-output quaternary chan- nel with quaternary sources X = Y = (0, 1,2,3), as defined by the following conditions on conditional probabilities: 13.3 (T): Determine the capacity of the four-input, eight-output channel characterized by the following transition matrix: 0 0 1 1 13.4 (M): Assuming an uncorrected bit-error-rate of BER 2 1 x 10-3, determine the corrected BER resulting from implementing the Hamming code (7, 4) in a binary channel. Is this code optimal in view of the channel coding theorem? 13.5 (B): Evaluate an upper bound for the corrected bit-error-rate when using the Reed-Solomon code RS(255,231) over a binary channel with noise parameter E=10-3 (clue: see text). Gaussian channel and Shannon-Hartley theorem This chapter considers the continuous-channel case represented by the Gaussian chan- nel, namely, a continuous communication channel with Gaussian additive noise. This will lead to a fundamental application of Shannon's coding theorem, referred to as the Shannon-Hartley theorem (SHT), another famous result of information theory, which also credits the earlier 1920 contribution of Ralph Hartley, who derived what remained known as the Hartley's law of communication channels. 1 This theorem relates channel capacity to the signal and noise powers, in a most elegant and simple formula. As a recent and little-noticed development in this field, I will describe the nonlinear channel, where the noise is also a function of the transmitted signal power, owing to channel nonlinearities (an exclusive feature of certain physical transmission pipes, such as opti- cal fibers). As we shall see, the modified SHT accounting for nonlinearity represents a major conceptual progress in information theory and its applications to optical com- munications, although its existence and consequences have, SO far, been overlooked in textbooks. This chapter completes our description of classical information theory, as resting on Shannon'