Coded trace reconstruction in a constant number of traces
Abstract
The \emph{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 blackbox reductions that show that averagecase 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 arXiv:
 arXiv:1908.03996
 Bibcode:
 2019arXiv190803996B
 Keywords:

 Computer Science  Information Theory;
 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics
 EPrint:
 31 pages, 2 figures, comments welcome