How can a $[8,4,4]_7$ linear code construct, a $[16,5,8]_2$ code?

Question

I'm given a code $[8,4,4]_7$ linear code how can I construct a $[16,5,8]_2$ code.

I am thinking $(u,u)$ construction and take a sub-code of it but that would give me a $d$ of 4 and not 8

Answer 1

Using $(u,u+v)$ construction will work, where $u=[8,4,4]_2, v=[8,1,8]_2$(a trivial code). In fact, $[16,5,8]_2$ is a Reed-Muller code $RM(1,4)$.