A sender is sending information bits in blocks of size K. The information goes through an error-prone channel to reach the receiver. An error correction/detection code is used at the transmitter and the receiver uses the corresponding decoder.

The encoder adds N redundancy bits to the K-bit long information , generating a codeword of length (K+N)-bit long.

The code can correct all single errors and detect up to 3 errors. The probability a bit is received in error due to channel impairments is P_B. The occurring bit error events are independent from each other, i.e. the event {bit i is in error} is independent of the event {bit j is in error}, where bit i and bit j belong to the same frame .

Calculate the probability:

(a) the decoder produces a correct output.

(b) The probability the decoder fails to correct errors or identify that errors have occurred.