A refinement of Cusick–Cheon bound for the second order binary Reed–Muller code

We prove a stronger form of the conjectured Cusick–Cheon lower bound for the number of quadratic balanced Boolean functions. We also prove various asymptotic results involving B ( k , m ) , the number of balanced Boolean functions of degree ≤ k in m variables, in the case k = 2 . Finally, we connect our results for k = 2 with the (still unproved) conjectures of Cusick–Cheon for the functions B ( k , m ) with k > 2 .