A Hamming (7,4 ) code uses the following check bit rulesc1 = k1 + k4 c2  = k1 + k2 + k3   c3 = k1 + k3Construct a complete table for the code set.Construct the Generator Matrix for the code setConstruct the Check Matrix for the code set.Using the Generator matrix, determine the encoded form of the data word <1001>Using the Check matrix decode the following received codeword <1001001>.

Sure, let's break this down step by step.

1. Construct a complete table for the code set:

We have 4 data bits (k1, k2, k3, k4) and 3 check bits (c1, c2, c3). The table would look like this:

2. Construct the Generator Matrix for the code set:

The generator matrix G is a 4x7 matrix, where the left 4x4 part is an identity matrix and the right 4x3 part is the transpose of the parity-check matrix.

G = [ 1 0 0 0 c1 0 1 0 0 c2 0 0 1 0 c3 0 0 0 1 c4 ]

3. Construct the Check Matrix for the code set:

The check matrix H is a 3x7 matrix, where the left 3x3 part is the transpose of the parity-check matrix and the right 3x4 part is an identity matrix.

H = [ c1 c2 c3 1 0 0 c1 0 c2 0 1 0 c1 c2 0 0 0 1 ]

4. Using the Generator matrix, determine the encoded form of the data word <1001>:

To encode the data word, we multiply it by the generator matrix.

Encoded word = Data word * G

5. Using the Check matrix decode the following received codeword <1001001>:

To decode the received codeword, we multiply it by the check matrix.

Decoded word = Received codeword * H

If the result is a zero vector, then the received codeword is error-free. If not, the result indicates the position of the error.