Polar Codes' Simplicity, Random Codes' Durability
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Over any discrete memoryless channel, we build codes such that: for one, their block error probabilities and code rates scale like random codes'; and for two, their encoding and decoding complexities scale like polar codes'. Quantitatively, for any constants π,ρ>0 such that π+2ρ<1, we construct a sequence of error correction codes with block length N approaching infinity, block error probability (-N^π), code rate N^-ρ less than the Shannon capacity, and encoding and decoding complexity O(Nlog N) per code block. The putative codes take uniform ς-ary messages for sender's choice of prime ς. The putative codes are optimal in the following manner: Should π+2ρ>1, no such codes exist for generic channels regardless of alphabet and complexity.
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