For each value of the parameters $A,n,d$, a linear program exists whose integer solutions correspond to codes. The Plotkin bound gives a necessary and sufficient condition on $n/d$ for feasibility. Some further simple remarks on the tableau of the linear program can be made; it can also be modified to consider only linear codes. For A divisible by 4, codes with optimum $n/d$ with smallest possible value for n are Hadamard matrices. The question of a bound on n is an instance of the more general question of an upper bound on b for a block design with given v and k, which is conjectured to be polynomial in v. Some constructions related to these questions are given.