My question is most precisely stated in the title. As an example, if we consider base 10, and k=4, then I am asking, is it possible to have a sequence of length 10^4 + 3, such that each 4 digit number appear exactly once in a consecutive block? This is related to finding a Hamiltonian cycle in a b-regular graph. But one would also like to construct the actual cycle. 



how to find a sequence of digits in base b such that each consecutive block of size k appears exactly once?
Google for "generating De Bruijn sequences".


