A composition of $n$ is a sequence of positive integers which sum to $n$. Such a sequence can be split into runs, where a run is a maximal contiguous subsequence of equal terms.

For example, $2,2,1,1,1,3,2,2$ is a composition of $14$ consisting of four runs:

$2,21,1,132,2$

Let $F(n)$ be the number of compositions of $n$ where every run has odd length.

For example, $F(5)=10$:

$$\begin{array}{rlrlrlrlrl}& 5& & 4,1& & 3,2& & 2,3& & 2,1,2\\ & 2,1,1,1& & 1,4& & 1,3,1& & 1,1,1,2& & 1,1,1,1,1\end{array}$$

Find $F({10}^5)$. Give your answer modulo $1111124111$.