Polar Codes' Simplicity, Random Codes' Durability

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 $\pi,\rho>0$ such that $\pi+2\rho 1$, no such codes exist for generic channels regardless of alphabet and complexity.

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