(2026-01) On those Boolean functions having only one Walsh zero
2026-01-09
Boolean functions having only one Walsh zero (or equivalently up to a translation, balanced functions whose sums with non-constant affine Boolean functions are all unbalanced) have been constructed for every n ≥ 10, by Mesnager and the first author, twenty years ago. This same paper had checked (partly mathematically and partly thanks to computer investigations) that no such function exists for n ≤ 6 but left open the question of constructing them for 7 ≤ n ≤ 9. Since then, functions in 7, 8 and 9 variables having one Walsh zero have been found by Lou and Wang, thanks to ad hoc methods combined with computer searches, but not as elements in infinite classes of functions having this property. In the present paper, we provide such infinite classes for n ≥ 8. For n = 7, we provide one more function (found by a computer investigation thanks to an algorithm) but we leave open the possibility of finding an infinite class valid for n ≥ 7. We also provide a secondary construction of functions with one Walsh zero in n + 2 variables from such functions in n variables, which does not need particular conditions on the latter for being successful (and which provides then a systematic way to obtain functions in n + 2 variables from functions in n variables). We investigate mathematical proofs of non-existence of such functions in n ≤ 6 variables.