Binary De Bruijn Cycles under Different Equivalence Relations

De Bruijn cycles are cyclic binary strings of length n where all substrings of length i are distinct. We present a generalization called (�, i)-De Bruijn cycles that are defined for an equivalence relation � on substrings of length i. In this paper, binary (�, i)-De Bruijn cycles under the equivalences of cyclic rotation, inverses, and flipping are examined. For the first two equivalences, we present exact solutions for all pairs (i, n) for which (�, i)-De Bruijn cycles of length n exist; for the last equivalence a conjecture is made about which pairs hold