Optimal Reconstruction Codes for Deletion Channels
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The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication scenario where the sender transmits a codeword from some codebook and the receiver obtains multiple noisy reads of the codeword. Motivated by modern storage devices, we introduced a variant of the problem where the number of noisy reads N is fixed (Kiah et al. 2020). Of significance, for the single-deletion channel, using log_2log_2 n +O(1) redundant bits, we designed a reconstruction code of length n that reconstructs codewords from two distinct noisy reads. In this work, we show that log_2log_2 n -O(1) redundant bits are necessary for such reconstruction codes, thereby, demonstrating the optimality of our previous construction. Furthermore, we show that these reconstruction codes can be used in t-deletion channels (with t> 2) to uniquely reconstruct codewords from n^t-1+O(n^t-2) distinct noisy reads.
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