  We study the size of the set of ergodic directions for the directional
billiard flows on the infinite band $\R\times [0,h]$ with periodically placed
linear barriers of length $0<\lambda<h$. We prove that the set of ergodic
directions is always uncountable. Moreover, if $\lambda/h\in(0,1)$ is rational
the Hausdorff dimension of the set of ergodic directions is greater than 1/2.
In both cases (rational and irrational) we construct explicitly some sets of
ergodic directions.
