An n-bit (cyclic) Gray code is a (cyclic) sequence of all n-bit strings such that consecutive strings differ in a single bit. We describe an algorithm which for every positive integer n constructs an n-bit cyclic Gray code whose graph of transitions is the d-dimensional hypercube Qd if n = 2^d, or a subgraph of Qd if 2^d−1 < n < 2^d. This allows to compress sequences that follow this code so that only Theta (log log n) bits per n-bit string are needed.