Consider a digital communication system that uses a repetition code for the channel encoding/decoding. In particular, each transmission is repeated n times, where n = 2m + 1 is an odd integer.
The decoder operates as follows. If in a block of n received bits the number of 0s exceeds the number of 1s, then the decoder decides in favor of a 0; otherwise, it decides in favor of a 1.
An error occurs when m + 1 or more transmissions out of n = 2m + 1 are incorrect. Assume a binary symmetric channel.
a. For n = 3, show that the average probability of error is given by
