The embedding of linearly ordered sets

It is shown that if a linearly ordered set $B$ does not contain as subsets sets of order type $\omega_\alpha$ and $\omega_\alpha^*$, then $B$ can be embedded in $2^{\omega_\alpha}$. We construct an example of a set satisfying the above conditions which cannot be embedded in any $2^\beta$ if $\beta<\omega_\alpha$. Simultaneously we show that for any ordinal $\alpha$, $2^{\alpha+1}$ cannot be embedded in $2^\alpha$ and that there exists at least $\chi_{\alpha+1}$ distinct dense order types of cardinality $2^{\chi_\alpha}$.