# Coded trace reconstruction in a constant number of traces

## TL;DR

This paper introduces efficient binary codes that can be reconstructed from a constant number of traces in the binary deletion channel, advancing the understanding of trace reconstruction with near-optimal rates and trace complexity.

## Contribution

It provides the first construction of binary codes with high rate that are recoverable from a constant number of traces, and establishes bounds relating average-case and coded trace reconstruction.

## Key findings

- Codes of rate 1−ε recoverable from a constant number of traces
- Lower bounds on the number of traces needed for high-rate codes
- Equivalence between average-case and coded trace reconstruction

## Abstract

The coded trace reconstruction problem asks to construct a code $C\subset \{0,1\}^n$ 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 ${\exp(O_q(\log^{1/3}(\frac{1}{\varepsilon})))}$ (a constant independent of $n$) traces of a $\operatorname{BDC}_q$ for any constant deletion probability $q\in(0,1)$. We also show that, for rate $1-\varepsilon$ binary codes, $\tilde \Omega(\log^{5/2}(1/\varepsilon))$ 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 $O_{\varepsilon}(1)$-sized alphabet that are recoverable from $O(\log(1/\varepsilon))$ traces, and that this is tight.