$4\mathrm{.}24($programming required$)$ Write a program, in a programming language of your choice, to verify that any two codewords in the Hamming code differ in at least three bit positions.

Let's find the "next" Hamming code, with $7-$bit messages and $11-$bit codewords and a minimum distance of $3.$ We'll use the same style of codeword as in Definition $4\mathrm{.}13$: the first $7$ bits of the codeword will simply be the message, and the next $4$ bits will be parity bits $($each for some subset of the message bits$).$