Linear time construction of a compressed Gray code

An n -bit (cyclic) Gray code is a (cyclic) ordering of all n -bit strings such that consecutive strings differ in exactly one bit. We construct an n -bit cyclic Gray code C n whose graph of transitions is isomorphic to an induced subgraph of the d -dimensional hypercube where d = ⌈ lg n ⌉ . This allows to represent C n so that only Θ ( log log n ) bits per n -bit string are needed. We provide an explicit description of an algorithm which generates the transition sequence of C n in linear time with respect to the output size.