In a digital communication system, a block of data bits is mapped into an - bit codeword that typically contains the information bits as well as n – k redundant bits. This is known as an (n, k) block code. The redundant bits are included to provide error correction capability. Suppose that each transmitted bit in our digital communication system is received in error with probability p. Furthermore, assume that the decoder is capable of correcting any pattern of t or fewer errors in an bit block. That is, if t or less bits in an n bit block are received in error, then the codeword will be decoded correctly, whereas if more than errors occur, the decoder will decode the received word incorrectly. Assuming each bit is received in error with probability p = 0.03, find the probability of decoder error for each of the following codes.
(a) (n, k) = (7, 4) t = 1
(b) (n, k) = (15, 7) t = 2
(c) (n, k) = (31, 16) t = 3