On the Existence of Balanced Generalized de Bruijn Sequences

Abstract

A balanced generalized de Bruijn sequence with parameters $(n,l,k)$ is a cyclic sequence of $n$ bits such that (a) the number of 0's equals the number of 1's, and (b) each substring of length $l$ occurs at most $k$ times. We determine necessary and sufficient conditions on $n,l$, and $k$ for the existence of such a sequence.