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 $\Theta(n)$ and a new construction that attains the previously known upper bound of $\Theta(\frac{2^n}{\sqrt{n}})$. This extends the work of Cooper and Heitsch~[\emph{Discrete Mathematics}, 310 (2010)].

翻译：二进制字符串的差异是指给定的二进制字符串中所有可能的子字符串中数量和零之间的最大(绝对)差异。我们提供了对已知的布鲁伊恩序列简单构造差异的调查。此外,我们演示了达到美元(n)的较低约束值的建筑工程和达到以前已知的美元($)的上限($)的新建筑。这延伸了库珀和海特施的作品[\emph{Discrete数学},310(2010年)]。