For a set of integers I, we define a q-ary I-cycle to be an assignment of the symbols 1 through q to the integers modulo q n so that every word appears on some translate of I. This definition generalizes that of de Bruijn cycles, and opens up a multitude of questions. We address the existence of such cycles, discuss “reduced ” cycles (ones in which the all-zeroes string need not appear), and provide general bounds on the shortest sequence which contains all words on some translate of I. We also prove a variant on recent results concerning decompositions of complete graphs into cycles and employ it to resolve the case of |I | = 2 completely
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