Lower bounds for trace reconstruction

Abstract: In the trace reconstruction problem, an unknown bit string ${\bf x}\in{0,1 }<sup>n$ is sent through a deletion channel where each bit is deleted independently with some probability $q\in(0,1)$, yielding a contracted string $\widetilde{\bf x}$. How many i.i.d.\ samples of $\widetilde{\bf x}$ are needed to reconstruct $\bf x$ with high probability? We prove that there exist ${\bf x},{\bf y} \in{0,1 }<sup>n$ such that at least $c\, n^{{5/4}/\sqrt{\log} n}$ traces are required to distinguish between ${\bf x}$ and ${\bf y}$ for some absolute constant $c$, improving the previous lower bound of $c\,n$. Furthermore, our result improves the previously known lower bound for reconstruction of random strings from $c \log² n$ to $c \log^{{9/4}n/\sqrt{\log} \log n}$.