Upper estimate of realization complexity of linear functions in a basis consisting of multi-input elements

Abstract

The paper is focused on realization of parity functions by circuits in the basis U _∞. This basis contains all functions of the form ${\left( {x_1^{\sigma 1}\& ...\& x_k^{\sigma k}} \right)^\beta }$. A method of construction of circuits for a parity function of n variables with the complexity $\left\lfloor {\left( {7n - 4} \right)/3} \right\rfloor$ is described. This improves the previously known upper bound of U _∞-complexity of parity functions that was $\left\lceil {\left( {5n - 1} \right)/2} \right\rceil$. The minimality of constructed circuits is verified for very small n (for n < 7).