Upper bounds on the Rate of Uniformly-Random Codes for the Deletion Channel
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We consider the maximum coding rate achievable by uniformly-random codes for the deletion channel. We prove an upper bound that's within 0.1 of the best known lower bounds for all values of the deletion probability d, and much closer for small and large d. We give simulation results which suggest that our upper bound is within 0.05 of the exact value for all d, and within 0.01 for d>0.75. Despite our upper bounds, based on simulations, we conjecture that a positive rate is achievable with uniformly-random codes for all deletion probabilities less than 1. Our results imply impossibility results for the (equivalent) problem of compression of i.i.d. sources correlated via the deletion channel, a relevant model for DNA storage.
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