Scott sentence complexities of linear orderings

We study possible Scott sentence complexities of linear orderings using two approaches. First, we investigate the effect of the Friedman-Stanley embedding on Scott sentence complexity and show that it only preserves ${\mathrm{\Pi }}_{\alpha }^{\mathrm{i}\mathrm{n}}$ complexities. We then take a more direct approach and exhibit linear orderings of all Scott complexities except ${\mathrm{\Sigma }}_3^{\mathrm{i}\mathrm{n}}$ and ${\mathrm{\Sigma }}_{\lambda +1}^{\mathrm{i}\mathrm{n}}$ for $\lambda$ a limit ordinal. We show that the former can not be the Scott sentence complexity of a linear ordering. In the process we develop new techniques which appear to be helpful to calculate the Scott sentence complexities of structures.