Consider the Hamming binary code Ham$(4,2).$

Write the parity check matrix H$\_4$ for this code.

If G is a generator matrix for this code, what is the size of G $($how many rows, how many columns$)?$

Determine the parameters $($n$,$k$,$d$)$ of this code. How many codewords are there in this code? How many errors can it detect? How many can it correct?

Use the parity check matrix H$\_4$ to decode the messages $111000000000111$ and $000111000111000.$