Construct an example of linear code given $(n,k,d)$

Question

I am trying to construct an example of a linear code $C$ given its length $n$, dimension $k$ and distance $d$.

Which is the most efficient way to find all the codewords of $C$ or alternately to construct its generating matrix $G$?

Answer 1

Things are not so simple. There is no general recipe. For the particular case $d=3$ you have Hamming codes (but the allowed values of $(n,k)$ are restricted). A more general recipe (allows for arbitrary $d$, but it's much more complex - and not necessarily binary) is the BCH code design - again the values of $(n,k)$ are restricted.