Sequence Reconstruction Problem for Deletion Channels: A Complete Asymptotic Solution
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Transmit a codeword x, that belongs to an (ℓ-1)-deletion-correcting code of length n, over a t-deletion channel for some 1≤ℓ≤ t<n. Levenshtein, in 2001, proposed the problem of determining N(n,ℓ,t)+1, the minimum number of distinct channel outputs required to uniquely reconstruct x. Prior to this work, N(n,ℓ,t) is known only when ℓ∈{1,2}. Here, we provide an asymptotically exact solution for all values of ℓ and t. Specifically, we show that N(n,ℓ,t)=2ℓℓ/(t-ℓ)! n^t-ℓ - O(n^t-ℓ-1) and in the special instance where ℓ=t, we show that N(n,ℓ,ℓ)=2ℓℓ. We also provide a conjecture on the exact value of N(n,ℓ,t) for all values of n, ℓ, and t.
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