All Integers from the Smallest Digit Stream with a Window Filter Let's represent integers with D digits where each digit has B values
(i.e., the base is B and we effectively work only with integers between
1 and B^D).  Is it possible to choose a single cyclic/repeating stream
of B^D digits and D relative position "windows" (to look into that
stream and filter out the digits we view) such that all integers come
out?
As an example, for D=3 and B=3, the stream "1 2 1 2 3 3 1 1 2 3 1 1 3
2 3 2 1 2 2 2 1 2 1 3 3 3 3" with window positions at "1 11 18" generates
the full range of integers exactly once per cycle (the first integer
generated is 112, then 212, ...).
I'm actually looking for an answer for D=4 and B=4 if there is no
general method.
 A: Yes. @GerhardPaseman's comment about De Bruijn cycles points to the solution:

a $k$-ary De Bruijn sequence $B(k, n)$ of order $n$ is a cyclic sequence of a given alphabet $A$ with size $k$ for which every possible subsequence of length $n$ in $A$ appears as a sequence of consecutive characters exactly once.

Now the result you require follows from the fact that

De Bruijn sequences of all orders exist.

The relevant reference is:

van Aardenne-Ehrenfest, T.; de Bruijn, N. G. (1951), Circuits and trees in oriented linear graphs, Simon Stevin 28: 203--217.

The special case $k=2$ was first proved here:

Flye Sainte-Marie, C. (1894), Solution to question nr. 48, L'intermédiaire des Mathématiciens 1: 107--110.

More details here (including the intriguing tidbit that "the earliest known example of a De Bruijn sequence comes from Sanskrit prosody"; make of that what you will).
A: The more generalized problem (given window positions, and alphabet size, find a cyclic or even non cyclic zequence such that shifting window positions gives all words of length same as concatenated window) sounds like a recurrence problem to me.  For window length of 2 with position difference k coprime to the alphabet size and possibly to 2, one can write the DeBruijin cycle in skip digit fashion.  So for window positions 1 and 4 and alphabet size coprime to 3 start filling in positions 1,4,7,10,13,... with consecutive members of the DeBruijn string.  Also one might check out gray coding systems to see if any resemble what bobuhito wants.
