Question $3.$

Consider a code of $10-$bits long words $($from $0000000000$ to $1111111111).$ Now, you add

$4$ parity bits, $p_1,p_2,p_3$ and $p_4$ for each word, with:

$p_1=b_{9}o+b_{8}o+b_{7}o+b_{6}o+b_5$

$p_2=b_{4}o+b_{3}o+b_{2}o+b_{1}o+b_0$

$p_3=p_{1}o+p_2$

$p_4=p_3$

What is the Hamming distance of this new code? Explain in a rigorous manner.