  A pruned variant of polar coding is reinvented for all binary erasure
channels. For small $\varepsilon>0$, we construct codes with block length
$\varepsilon^{-5}$, code rate $\text{Capacity}-\varepsilon$, error probability
$\varepsilon$, and encoding and decoding time complexity
$O(N\log|\log\varepsilon|)$ per block, equivalently $O(\log|\log\varepsilon|)$
per information bit (Propositions 5 to 8).
  This result also follows if one applies systematic polar coding [Ar{\i}kan
10.1109/LCOMM.2011.061611.110862] with simplified successive cancelation
decoding [Alamdar-Yazdi-Kschischang 10.1109/LCOMM.2011.101811.111480], and then
analyzes the performance using [Guruswami-Xia arXiv:1304.4321] or
[Mondelli-Hassani-Urbanke arXiv:1501.02444].
