In a network with capacity h for multicast, information X^h=(X₁, X₂, …, X_h) can be transmitted from a source node to sink nodes without error by a linear network code. Furthermore, secret information S^r=(S₁, S₂, …, S_r) can be transmitted securely against wiretappers by k-secure network coding for k≤h-r. In this case, no information of the secret leaks out even if an adversary wiretaps k edges, i. e. channels. However, if an adversary wiretaps k+1 edges, some S_i may leak out explicitly. In this paper, we propose strongly k-secure network coding based on strongly secure ramp secret sharing schemes. In this coding, no information leaks out for every (S_i1, S_i2, …,S_ir-j) even if an adversary wiretaps k+j channels. We also give an algorithm to construct a strongly k-secure network code directly and a transform to convert a nonsecure network code to a strongly k-secure network code. Furthermore, some sufficient conditions of alphabet size to realize the strongly k-secure network coding are derived for the case of k<h-r.