De Bruijn Sequences with Minimum Discrepancy

Abstract

The discrepancy of a binary string is the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. In this note we determine the minimal discrepancy that a binary de Bruijn sequence of order $n$ can achieve, which is $n$. This was an open problem until now. We give an algorithm that constructs a binary de Bruijn sequence with minimal discrepancy. A slight modification of this algorithm deals with arbitrary alphabets and yields de Bruijn sequences of order $n$ with discrepancy at most $1$ above the trivial lower bound $n$.