1. Hamming codes are said to be perfect single-error correcting codes. Justify the fact that Hamming codes are perfect.

2. Consider the following statement:
Explain the conditions under which this statement is justified.

3. In a repetition code, a single message bit is encoded into a block of identical bits to produce an (n, 1). Considering the (5, 1) repetition code, evaluate the syndrome for:

a. All five possible single-error patterns.

b. All ten possible double-error patterns.