Log-Logarithmic Time Pruned Polar Coding

A pruned variant of polar coding is proposed for binary erasure channel (BEC). Fix any BEC. For sufficiently small ε > 0, we construct a series of capacity achieving codes with block length N = ε-4.9, code rate R = Capacity - O(ε), block error probability P = ε, and encoding and decoding time complexity bC = O(log|log ε|) per information bit. The given per-bit complexity bC is log-logarithmic in N, in Capacity - R, and in P. Beyond BEC, there is a generalization: Fix a prime q and fix a symmetric, q-ary-input, discrete-output memoryless channel. For sufficiently small ε > 0, we construct a series of error correction codes with block length N = ε-constant, code rate R = Capacity - O(ε), block error probability P = ε, and encoding and decoding time complexity bC = O(log|log ε|) per information bit. Over general channels, this family of codes has the lowest per-bit time complexity among all capacity-achieving codes known to date.