Coded trace reconstruction in a constant number of traces
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The coded trace reconstruction problem asks to construct a code C⊂{0,1}^n such that any x∈ C is recoverable from independent outputs ("traces") of x from a binary deletion channel (BDC). We present binary codes of rate 1-ε that are efficiently recoverable from (O_q(log^1/3(1/ε))) (a constant independent of n) traces of a BDC_q for any constant deletion probability q∈(0,1). We also show that, for rate 1-ε binary codes, Ω̃(log^5/2(1/ε)) 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-ε over an O_ε(1)-sized alphabet that are recoverable from O(log(1/ε)) traces, and that this is tight.
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