The general problem:

Determine all triples [n,k,d] for which there exists a binary linear code of length n, dimension k, and minimum distance d.

The Delsarte-MacWilliams method for obtaining nonexistence results:

For a hypothetical [n,k,d] code C, let denote its number of words of weight i. Let denote its dual code. Then is a linear combination of .

Since for all i, we obtain a system of inequalities which must have a solution in order for C to exist. Further, by linear programming, we may (sometimes) show that the system has no solution.