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)B(k,m), the number of balanced Boolean functions of degree ≤k≤k in mm variables, in the case k=2k=2. Finally, we connect our results for k=2k=2 with the (still unproved) conjectures of Cusick–Cheon for the functions B(k,m)B(k,m) with k>2k>2.