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Let<tex>$N(d,d^perp)$</tex>denote the minimum length<tex>$n$</tex>of a linear code<tex>$C$</tex>with<tex>$d$</tex>and<tex>$d^bot$</tex>, where<tex>$d$</tex>is the minimum Hamming distance of<tex>$C$</tex>and<tex>$d^bot$</tex>is the minimum Hamming distance of<tex>$C^bot$</tex>. In this correspondence, we show lower bounds and an upper bound on<tex>$N(d,d^perp)$</tex>. Further, for small values of<tex>$d$</tex>and<tex>$d^perp$</tex>, we determine<tex>$N(d,d^perp)$</tex>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

Primal-Dual Distance Bounds of Linear Codes With Application to Cryptography