A note on the FMEI of the Boolean functions in the Generalized Maiorana-McFarland construction

This paper investigates well-known conjectures in Boolean function analysis, specifically focusing on the Fourier Min-Entropy/Influence (FMEI) conjecture, a natural relaxation of the more established Fourier Entropy/Influence (FEI) conjecture. While the FEI conjecture proposes that the Fourier entropy of a Boolean function is bounded by a constant multiple of its total influence, the FMEI conjecture substitutes entropy with min-entropy. We present a construction of Boolean functions that establishes a new lower bound on the universal constant of the FMEI conjecture. By employing the Generalized Maiorana-McFarland construction with suitably chosen injective mappings, we construct Boolean functions whose FMEI value surpasses the previous bound 2.8444. Specifically, our construction can yield functions demonstrating an FMEI value strictly less than 4 but arbitrarily close to 4, and we provide the conditions for the FMEI value to exceed 3. Furthermore, we investigate other classes of plateaued functions, such as partially bent functions and the Address functions, and prove that relaxing the injective constraint in the Generalized Maiorana-McFarland construction cannot increase the FMEI value. Thereby, it provides new insights toward understanding of the FMEI conjecture.