The bad science matrix problem consists in finding, among all matrices $A \in \mathbb{R}^{n \times n}$ with rows having unit $\ell^2$ norm, one that maximizes $\beta(A) = \frac{1}{2^n} \sum_{x \in \{-1, 1\}^n} \|Ax\|_\infty$. Our main contribution is an explicit construction of an $n \times n$ matrix $A$ showing that $\beta(A) \geq \sqrt{\log_2(n+1)}$, which is only 18% smaller than the asymptotic rate. We prove that every entry of any optimal matrix is a square root of a rational number, and we find provably optimal matrices for $n \leq 4$.

翻译：伪科学矩阵问题旨在从所有行向量具有单位$\ell^2$范数的矩阵$A \in \mathbb{R}^{n \times n}$中，找到一个最大化$\beta(A) = \frac{1}{2^n} \sum_{x \in \{-1, 1\}^n} \|Ax\|_\infty$的矩阵。我们的主要贡献是显式构造了一个$n \times n$矩阵$A$，证明$\beta(A) \geq \sqrt{\log_2(n+1)}$，该值仅比渐近速率低18%。我们证明了任何最优矩阵的每个元素都是有理数的平方根，并找到了$n \leq 4$时可证明最优的矩阵。