Hamming Code C(7, 4)

We consider a multiple parity detecting and correcting code such that: k =

4 and n = 7. The bits of the codeword y are obtained from the word to be coded u in accordance with the following parity relations:

YI = UI E9 U2 E9 U4, Y2 = UI Ea Uj Ea U4, Yj = UI, Y4 = U2 E9 uj E9 U4, Y5 = U2, Y6 = Uj, Y7 = U4, where Uj and Yi are the bits i of u and y.

1. Analyze this code and show that it detects all single errors and all double errors.

2. Show that this code only detects and corrects all single errors.

3. The definition of this code corresponds to a simple exchange of the relations given for the code of Example 15-4. Compare the detecting and correcting capabilities of these two codes.

4. How can we modify this code so that it is able to detect all single and double errors AND correct all single errors (without making any confusion between them)?