Investigating the discrepancy property of de Bruijn sequences

The discrepancy of a binary string refers to the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. We provide an investigation of the discrepancy of known simple constructions of de Bruijn sequences. Furthermore, we demonstrate constructions that attain the lower bound of $\mathrm{\Theta }(n)$ and a new construction that attains the previously known upper bound of $\mathrm{\Theta }(\frac{2^n}{\sqrt{n}})$. This extends the work of Cooper and Heitsch~[\emph{Discrete Mathematics}, 310 (2010)].