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

Let N(d,d perp ) denote the minimum length n of a linear code C with d and d perp , where d is the minimum Hamming distance of C and d perp is the minimum Hamming distance of C perp . 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 et al