The Fourier entropy of linear monotone DNFs on few variables

A monotone DNF is linear if any two of its terms share at most one variable. Over all linear monotone DNFs with at most eight relevant variables, of which there are 552,251 isomorphism classes on exactly eight, we determine by exhaustive enumeration with certified logarithms the largest value of the ratio $H(f)/I(f)$ of Fourier entropy to total influence and the largest value of minᵢEᵢ(f)/Infᵢ(f), where Eᵢ is the erasure entropy at i. They are 3.8891054198… and 3.3047013224…, and each is attained by a single class. A transversal design on $21$ variables has H/I=4.1346…, so neither value is the supremum over the class. We then give three sufficient conditions for a coordinate with Eᵢ≤ C\,Infᵢ in a linear monotone DNF: a load certificate that assumes no bound on widths or degrees, a criterion that involves the loads of the variables of a single term only, and a bound in terms of the bias of f. On the affine transversal designs with k+2log₂k+1 groups of 2ᵏ points each, all of these conditions fail at every coordinate, as do the width and variance bounds of earlier work, while [F=0] tends to zero more slowly than any exponential in the number of variables.