On the Structure of Bad Science Matrices

The bad science matrix problem consists in finding, among all matrices A ∈ Rn × n with rows having unit ² norm, one that maximizes β(A) = 1/2ⁿ ∑_x ∈ \-1, 1\ⁿ \|Ax\|_∞. Our main contribution is an explicit construction of an n × n matrix A showing that β(A) ≥ √log₂(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 ≤ 4.