Errors occurring during data transmission is a simple reality of digital communications. Thus it has become important to be able to detect and also to correct errors. A repetition code is a transmission scheme in which a given digital sequence is transmitted multiple times, and a majority consensus is used to determine the final sequence which is accepted.

For example, consider a 4-bit binary sequence 1011. In a 3-repetition scheme, this sequence is transmitted by the sender three times through a potentially noisy communication channel. Suppose that a recipient observes the sequence: 1011 1111 1010. Notice that the first repetition is transmitted correctly, but there is a single error in each of the second and third repetitions. Applying a majority consensus at each bit position, the recipient is able to reconstruct the original sequence 1011.

In this question, you will construct a simple circuit to apply a consensus correction for a 3-repetition error-correction scheme.

Complete the truth table for obtaining a consensus from three individual bits representing one bit of three repetitions of a received sequence:

r1 r2 r3 maj

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

From your truth table above, determine a propositional logic expression for maj:

maj=