On the complexity of the realization of a linear function by formulas in finite Boolean bases

We completely describe the set of bases over which the complexity of realization of the function $x_1\oplus\ldots\oplus x_n$ is of order $n$. For all bases not belonging to this set, we obtain the lower bound for the complexity of realization of the function $x_1\oplus\ldots\oplus x_n$, which is of the form $n^c$, where $c>1$ and $c$ does not depend on $n$. Basing on this bound for complexity, we give a more simple proof of existence of an infinite (descending) sequence of Boolean bases.
The research was supported by the Russian Foundation for Basic Research, grant 99–01–01175, and also by FTP ‘Integration’, grant 473.