Polar Codes’ Simplicity, Random Codes’ Durability

Over any discrete memoryless channel, we offer error correction 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 π +2ρ <; 1, we construct a sequence of block codes with block length $N approaching infinity, block error probability exp (-N
^π
) , code rate N
^-ρ
less than the Shannon capacity, and encoding and decoding complexity O(N log N}) per code block. The core theme is to incorporate polar coding (which limits the complexity to polar's realm) with large, random, dynamic kernels (which boosts the performance to random's realm). The putative codes are optimal in the following manner: Should π +2ρ>1 , no such codes exist over generic channels regardless of complexity.