If you just want the optimal binary code for a certain set of parameters, then the easiest way would be to track down the references from that website and get ahold of the actual articles/books (or perhaps the authors). ;)

If you really want to do be able to do it computationally, this seems exceedingly difficult. One possible simplifying idea that I can suggest is to only look for linear codes. This will add some much-needed structure to the search space to exploit. You "only" need to search for a good parity-check matrix for the code. However, this might still be pretty computationally intensive. And since the numbers in that table are mostly powers of two, it is at least plausible that a linear code (whose number of codewords is always a power of two) can achieve the optimal parameters.