Liu–Mesnager–Chen upper-bound conjecture for distance to affine functions

Prove or refute that every map f:GF(2)^n\to GF(2)^m satisfies A_f\leq (1-2^{-m})(2^n-2^{n/2}), and, in the case n=m, that A_f\leq 2^n-2^{n/2}-1.

Background

The paper studies A_f, the minimum Hamming distance between a vectorial Boolean function f:GF(2)^n\to GF(2)^m and the set of affine maps. Liu, Mesnager, and Chen proposed a universal upper bound for this parameter. If valid, the bound would determine the exact distance for vectorial bent functions by matching the lower bound of Carlet, Ding, and Yuan.

The paper proves the conjectured bound for two classes of vectorial bent functions constructed from left pre-quasifields and for two specified monomial functions. However, the authors also report that the conjecture fails for the APN examples considered in dimensions 8 and 9, so the universal assertion remains unresolved as a general statement.