  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 paper, we
show a lower bound 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 et al.
