Primal-dual distance bounds of linear codes with application to cryptography

Let N(d, d(perpendicular to)) denote the minimum length a. of a linear code C with d and d(perpendicular to), where d is the minimum Hamming distance of C and d(perpendicular to) is the minimum Hamming distance of C-perpendicular to. In this correspondence, we show lower bounds and an upper bound on N(d, d(perpendicular to)). Further, for small values of d and d(perpendicular to), we determine N(d, d(perpendicular to)) 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.