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 Arikan 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 Arikan 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 0.1dB--0.25dB while maintaining the same decoding time. Moreover, ABS+ polar codes improve upon standard polar codes by 0.2dB--0.45dB 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