Building UV completions of lepton-number-violating effective operators has proved to be a useful way of studying and classifying models of Majorana neutrino mass. In this paper we describe and implement an algorithm that systematises this model-building procedure. We use the algorithm to generate computational representations of all of the tree-level completions of the operators up to and including mass-dimension 11. Almost all of these correspond to models of radiative neutrino mass. Our work includes operators involving derivatives, updated estimates for the bounds on the new-physics scale associated with each operator, an analysis of various features of the models, and a look at some examples. We find that a number of operators do not admit any completions not also generating lower-dimensional operators or larger contributions to the neutrino mass, ruling them out as playing a dominant role in the neutrino-mass generation. Additionally, we show that there are at most five models containing three or fewer exotic multiplets that predict new physics that must lie below 100 TeV. Accompanying this work we also make available a searchable database containing all of our results and the code used to find the completions. We emphasise that our methods extend beyond the study of neutrino-mass models, and may be useful for generating completions of high-dimensional operators in other effective field theories.
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