NP-hardness of minimum distance over a code
Berlekamp, McEliece and Van Tilborg 
[On the inherent intractability of certain coding problems, IEEE Trans. Information Theory, 24 (1978)]
proved that deciding if a binary code contains a code word of weight $w$ is NP-complete. In http://statweb.stanford.edu/~cgates/PERSI/papers/85_04_radon.pdf
Diaconis and Graham prove that deciding if a binary code of length $2m$ contains a code word of weight $n$ is NP-complete.