Geodesic
Minimal contact circuits for characteristic functions of spheres
Diskretnaya Matematika, Tome 32 (2020) no. 3, pp. 68-75.

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We study the complexity of implementation of the characteristic functions of spheres by contact circuits. By the characteristic functions of the sphere with center at a vertex $\tilde\sigma=(\sigma_1,\ldots,\sigma_n)$, $\sigma_1,\ldots,\sigma_n\in\{0,1\}$, we mean the Boolean function $\varphi^{(n)}_{\tilde\sigma}(x_1,\ldots,x_n)$ which is equal to 1 on those and only those tuples of values that differ from the tuple $\tilde\sigma$ only in one digit. It is shown that the number $3n-2$ of contacts is necessary and sufficient for implementation of $\varphi^{(n)}_{\tilde\sigma}(\tilde x)$ by a contact circuit.
Keywords: Boolean function, contact circuit, minimal circuit. 
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N. P. Red'kin. Minimal contact circuits for characteristic functions of spheres. Diskretnaya Matematika, Tome 32 (2020) no. 3, pp. 68-75. http://geodesic.mathdoc.fr/item/DM_2020_32_3_a4/

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