Lower bounds for trace reconstruction

08/04/2018

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by   Nina Holden, et al.

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In the trace reconstruction problem, an unknown bit string x∈{0,1}^n is sent through a deletion channel where each bit is deleted independently with some probability q∈(0,1), yielding a contracted string x. How many i.i.d. samples of x are needed to reconstruct x with high probability? We prove that there exist x, y∈{0,1 }^n such that we need at least c n^5/4/√( n) traces to distinguish between x and 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 ^2 n to c ^9/4n/√( n) .