Binary Hamming code

1-Define the product, , of two binary vectors of the same length to be the vector whose ith component is the product of the ith components of and .
Show that wt ( + ) =wt ( ) +wt ( ) – 2wt ( ) . wt means weight.

2. Suppose that a binary Hamming code is modified by adding an additional check bit to each codeword. This additional check bit is chosen so that each resulting vector has even weight.

a) Show that this procedure yields a linear code. You can use problem 1 to solve this part.
b) What are the parameter [n, k, d] of this new code? Prove your assertions, in particular for d.
c) How many errors can this new code correct?
d) How many errors can this new code detect, but not necessarily correct?
Note: This new code is called the extended Hamming code. Can you explain this definition

Can you give a specific example to understand this problem.

Please see the attached file for the fully formatted problems.