A (6, 3) systematic linear block code encodes the information sequence x = (x 1 , x 2 , x 3 )...

A (6, 3) systematic linear block code encodes the information sequence x = (x1, x2, x3) into codeword c = (c1, c2, c3, c4, c5, c6), such that c4 is a parity check on c1 and c2, to make the overall parity even (i.e., c1 ⊕ c2 ⊕ c4 = 0). Similarly c5 is a parity check on c2 and c3, and c6 is a parity check on c1 and c3.

1. Determine the generator matrix of this code.

2. Find the parity check matrix for this code.

3. Using the parity check matrix, determine the minimum distance of this code.

4. How many errors is this code capable of correcting?

5. If the received sequence (using hard decision decoding) is y = 100000, what is the transmitted sequence using a maximum-likelihood decoder? (Assume that the crossover probability of the channel is less than ½.)