Geodesic
On the number of perfectly balanced Boolean functions with barrier of length~$3$
Prikladnaâ diskretnaâ matematika, no. 1 (2011), pp. 26-33

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Some lower and upper bounds are obtained for the logarithm of the number of Boolean functions with the right barrier of length $3$ essentially depended on the last variable. Also, the following new lower bound for the logarithm of the number of perfectly balanced Boolean functions of $n$ variables with nonlinear dependence on the first and on the last variable is obtained: $2^{n-2}\left(1+\dfrac{\log_25}4-\mathrm O(1/\sqrt n)\right)$.
Keywords: perfectly balanced functions, barriers of Boolean functions, cryptography. 
@article{PDM_2011_1_a2,
     author = {S. V. Smyshlyaev},
     title = {On the number of perfectly balanced {Boolean} functions with barrier of length~$3$},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {26--33},
     publisher = {mathdoc},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2011_1_a2/}
}
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S. V. Smyshlyaev. On the number of perfectly balanced Boolean functions with barrier of length~$3$. Prikladnaâ diskretnaâ matematika, no. 1 (2011), pp. 26-33. http://geodesic.mathdoc.fr/item/PDM_2011_1_a2/