# Linear circuit complexity classes

The class $$\textrm{NC}^i$$ is the class functions computable by circuits families of bounded fan-in, $$n^{O(1)}$$ size and $$O(\log^i(n))$$ depth. The $$\textrm{NC}$$-hierarchy is the union of those classes.

Is there any study of the linear-size variant of this hierarchy? That is circuits families of bounded fan-in, polylog depth and linear size?

I know their exists some work with linear-$$\textrm{AC}^0$$ but nothing else. Remark that at least linear-$$\textrm{NC}^1$$ is nontrivial since it contains regular languages (and thus some $$\textrm{NC}^1$$-complete languages).

It follows from work of Valiant [1, 2] that linear-size $$\textrm{NC}^1$$ can be simulated by $$2^{O(n / \log \log n)}$$-size circuits of depth three and unbounded fan-in.