Very noisy channels, reliability functions, and exponentially optimum codes

A very noisy channel (VNC), is a discrete-input memoryless channel whose capacity is close to zero. Very noisy channels are of interest, since they serve as models for some important physical channels. There are two distinct classes of VNCs: Reiffen´s class I (Reiffen, 1963) and Majani´s class II (Majani, 1987). It is shown that the reliability function is known exactly for both classes of VNCs, by extending the results previously obtained only for class I. It is then shown that an exponentially optimum code can be constructed for a channel if it can be modeled as repeated uses of a binary-input class I VNC, a binary-input/binary-output class II VNC or a class II very noisy binary erasure channel. The theory developed is illustrated by considering the direct detection optical channel used with polarization modulation. The capacity, reliability function, and exponentially optimum code, for this channel, are derived