ABS+ Polar Codes: Exploiting More Linear Transforms on Adjacent Bits

ABS polar codes were recently proposed to speed up polarization by swapping certain pairs of adjacent bits after each layer of polar transform. In this paper, we observe that applying the Arıkan transform
$(U_{i}, U_{i+1}) \mapsto (U_{i}+U_{i+1}, U_{i+1})$
on certain pairs of adjacent bits after each polar transform layer leads to even faster polarization. In light of this, we propose ABS+ polar codes which incorporate the Arıkan transform in addition to the swapping transform in ABS polar codes. In order to efficiently construct and decode ABS+ polar codes, we derive a new recursive relation between the joint distributions of adjacent bits through different layers of polar transforms. Simulation results over a wide range of parameters show that the CRC-aided SCL decoder of ABS+ polar codes improves upon that of ABS polar codes by
$0.1 \mathop {\mathrm {dB}}\nolimits $
–
$0.25 \mathop {\mathrm {dB}}\nolimits $
while maintaining the same decoding time. Moreover, ABS+ polar codes improve upon standard polar codes by
$0.2 \mathop {\mathrm {dB}}\nolimits $
–
$0.45 \mathop {\mathrm {dB}}\nolimits $
when they both use the CRC-aided SCL decoder with list size 32. The implementations of all the algorithms in this paper are available at
https://github.com/PlumJelly/ABS-Polar