Construction of $n$ -Variable ( $n\equiv 2 \bmod 4$ ) Balanced Boolean Functions With Maximum Absolute Value in Autocorrelation Spectra $< 2^{\frac {n}2}$

In this paper, we consider the maximum absolute value Δ
_f
in the autocorrelation spectrum (not considering the zero point) of a function f. In an even number of variables n, bent functions possess the highest nonlinearity with Δ
_f
= 0. The long standing open question (for two decades) in this area is to obtain a theoretical construction of balanced functions with Δ
_f
<; 2
^n/2
. So far, there are only a few examples of such functions for n = 10, 14, but no general construction technique is known. In this paper, we mathematically construct an infinite class of balanced Boolean functions on n variables having absolute indicator strictly lesser than δ
_n
= 2
_n/2
- 2(
^(n+6)/4)
, nonlinearity strictly greater than ρ
_n
= 2
^n-1
-2
^n/2
+2
^n/2-3
-5·2
^((n-2)/4)
and algebraic degree n - 1, where n ≡ 2 (mod 4) and n ≥ 46. While the bound n ≥ 46 is required for proving the generic result, our construction starts from n = 18, and we could obtain balanced functions with Δ
_f
<; 2
^n/2
and nonlinearity > 2
^n-1
- 2
^n/2
for n = 18, 22, and 26.