Let$N(d,d^perp)$denote the minimum length$n$of a linear code$C$with$d$and$d^bot$, where$d$is the minimum Hamming distance of$C$and$d^bot$is the minimum Hamming distance of$C^bot$. In this correspondence, we show lower bounds and an upper bound on$N(d,d^perp)$. Further, for small values of$d$and$d^perp$, we determine$N(d,d^perp)$and give a generator matrix of the optimum linear code. This problem is directly related to the design method of cryptographic Boolean functions suggested by Kurosawa