kernels. In particular, we extend the special nodes of BPCs to their non-binary counterpart and define various non-binary special nodes in the SC decoding tree of NBPCs and propose their fast decoding. This way, we avoid traversing the full decoding tree and significantly reduce the decoding delay compared to symbol-by-symbol SC decoding. We also propose a simplified NBPC structure that facilitates the procedure of non-binary fast SC decoding. Using our proposed fast non-binary decoder, we observe an improvement of more than 96% in latency concerning the original SC decoding. This is while our proposed fast SC decoder for NBPCs incurs no error-rate loss.

DOI: 10.1109/tcomm.2024.3447781

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2\times 2

A method for efficiently successive cancellation (SC) decoding of polar codes with high-dimensional linear binary kernels (HDLBK) is presented and analyzed. We devise a

l

-expressions method which can obtain simplified recursive formulas of SC decoder in likelihood ratio form for arbitrary linear binary kernels to reduce the complexity of corresponding SC decoder. By considering the bit-channel transition probabilities

W_{G}^{(\cdot)}(\cdot|0)

and

W_{G}^{(\cdot)}(\cdot|1)

separately, a

W

-expressions method is proposed to further reduce the complexity of HDLBK based SC decoder. For a

m\times m

binary kernel, the complexity of straightforward SC decoder is

O(2^{m}N\log N)

. With

W

-expressions, we reduce the complexity of straightforward SC decoder to

O(m^{2}N\log N)

when

m\leq 16

. Simulation results show that

16\times16

kernel polar codes offer significant advantages in terms of error performances compared with

2\times2

kernel polar codes under SC and list SC decoders.

Construction and Decoding Algorithms for for Polar Codes based on

2\times2

Non-Binary Kernels

Polar codes based on

2\times2

non-binary kernels are discussed in this work. The kernel over

\text{GF}(q)

is selected by maximizing the polarization effect and using Monte-Carlo simulation. Belief propagation (BP) and successive cancellation (SC) based decoding algorithms are extended to non-binary codes. Additionally, a successive cancellation list (SCL) decoding with a pruned tree is proposed. Simulation results show that the proposed decoder performs very close to a conventional SCL decoder with significantly lower complexity.