A novel approach to the design of solid state nuclear magnetic resonance multiple pulse experiments is described. Based on time dependent perturbation theory the pulse cycle decoupling principle is extended to fifth order. Furthermore, by analyzing symmetry and commutator relations for high order terms in the average Hamiltonian expansion we introduce so-called z-rotational decoupling accomplished by concatenation of phase shifted pulse cycles. These fundamental tools prove extremely useful for the development of multiple pulse techniques capable of eliminating undesired interactions to high order in the average Hamiltonian expansion. The applicability of the methods is demonstrated by construction of homonuclear multiple-pulse decoupling methods which suppress pure dipolar terms up to fifth order and cross terms between rf inhomogeneity and dipolar coupling to second order in the Magnus expansion. For the dipolar terms this represents an improvement by two orders of magnitude compared to previous homonuclear decoupling sequences. High order truncation decoupling sequences based on the BLEW-12 and magic sandwich pulse cycles are compared to state-of-the-art methods numerically and by preliminary experiments.