I have been reading about residual codes, and showing how to improve upper bounds on the number of codewords for certain types of codes. I've come across one problem that I am having trouble with.
It starts with showing that $B_2(13, 6) \leq 2^4$ by using redisual codes, where $B_2(13, 6)$ is the number of binary linear codewords of length $13$ with a minimum distance of $6$.
Then, it asks to construct a linear binary code that meets this bound.
I am able to show $B_2(13, 6) \leq 2^4$, but I'm having trouble constructing such a code that meets this bound. Can anyone help with this construction? Thanks in advance.