LetCbe a binary code containing the zero word, with fundamental parametersd,s,d', ands, and lettbe a nonnegative integer such that the words of any fixed weight ofCform at-design, withtmaximal in this respect. It is shown thatt \geq \max(d-s',d'-s). Moreover, ifCcontains the all one word, thent \geq d'- s+1and ifCis an even weight code thent \geq d - s' + 1. Finally, the following inequality is derived:d'-s < d, provided thatd>2.