Coded trace reconstruction in a constant number of traces

Abstract: The coded trace reconstruction problem asks to construct a code $C\subset {0,1}<sup>n$ such that any $x\in C$ is recoverable from independent outputs ("traces") of $x$ from a binary deletion channel (BDC). We present binary codes of rate $1-\varepsilon$ that are efficiently recoverable from ${\exp(O_q(\log<sup>{1/3}(\frac{1}{\varepsilon})))}$ (a constant independent of $n$) traces of a $\operatorname{BDC}<em>q$ for any constant deletion probability $q\in(0,1)$. We also show that, for rate $1-\varepsilon$ binary codes, $\tilde \Omega(\log<sup>{5/2}(1/\varepsilon))$ traces are required. The results follow from a pair of black-box reductions that show that average-case trace reconstruction is essentially equivalent to coded trace reconstruction. We also show that there exist codes of rate $1-\varepsilon$ over an $O</em>{\varepsilon}(1)$-sized alphabet that are recoverable from $O(\log(1/\varepsilon))$ traces, and that this is tight.