List Decoding of Arıkan’s PAC Codes

Abstract

Polar coding gives rise to the first explicit family of codes that provably achieve capacity with efficient encoding and decoding for a wide range of channels. However, its performance at short blocklengths under standard successive cancellation decoding is far from optimal. A well-known way to improve the performance of polar codes at short blocklengths is CRC precoding followed by successive-cancellation list decoding. This approach, along with various refinements thereof, has largely remained the state of the art in polar coding since it was introduced in 2011. Recently, Arıkan presented a new polar coding scheme, which he called polarization-adjusted convolutional (PAC) codes. At short blocklengths, such codes offer a dramatic improvement in performance as compared to CRC-aided list decoding of conventional polar codes. PAC codes are based primarily upon the following main ideas: replacing CRC codes with convolutional precoding (under appropriate rate profiling) and replacing list decoding by sequential decoding. One of our primary goals in this paper is to answer the following question: is sequential decoding essential for the superior performance of PAC codes? We show that similar performance can be achieved using list decoding when the list size L is moderately large (say, L⩾128). List decoding has distinct advantages over sequential decoding in certain scenarios, such as low-SNR regimes or situations where the worst-case complexity/latency is the primary constraint. Another objective is to provide some insights into the remarkable performance of PAC codes. We first observe that both sequential decoding and list decoding of PAC codes closely match ML decoding thereof. We then estimate the number of low weight codewords in PAC codes, and use these estimates to approximate the union bound on their performance. These results indicate that PAC codes are superior to both polar codes and Reed–Muller codes. We also consider random time-varying convolutional precoding for PAC codes, and observe that this scheme achieves the same superior performance with constraint length as low as ν=2.

A Tree Pruning Technique for Decoding Complexity Reduction of Polar Codes and PAC Codes

“…While using a sequential decoding algorithm, the errorcorrection performance of the newly introduced polarizationadjusted convolutional (PAC) codes approaches the theoretical bounds with a very low computational complexity at high SNR values [4], [7]- [10]. Additionally, practical results demonstrate that when the list size L is somewhat large, equivalent errorcorrection performance can be attained using list decoding algorithm [11]. We also demonstrate the effectiveness of our suggested algorithm on the list decoding of PAC codes.…”

“…Similar to the SCL decoding of polar codes, the bit-metric function for the SCL decoding of PAC codes can be represented as (10) where the ith bit u i in the PAC code case is the input of the polar mapper. One distinction from decoding polar codes is that for the L paths of the list, L vectors (auxiliary shift registers) with sizes equal to the constraint length of the convolutional code must be created to compute u i [11]. To acquire L u i s for (11), a 0 should be placed into each shift register if it corresponds to a frozen bit or a zero branch; otherwise, a 1 should be loaded into each shift register.…”

“…One distinction from decoding polar codes is that for the L paths of the list, L vectors (auxiliary shift registers) with sizes equal to the constraint length of the convolutional code must be created to compute u i [11]. To acquire L u i s for (11), a 0 should be placed into each shift register if it corresponds to a frozen bit or a zero branch; otherwise, a 1 should be loaded into each shift register.…”

Achieving PAC Code Performance with SCL Decoding without Extra Computational Complexity

“…As stated in [19], the difference of PAC encoding scheme from standard polar encoding is the rate-1 convolutional encoder that takes place before polar transformation, which is implemented through a length-m memory as follows [19]:…”

CRC-Aided Adaptive BP Decoding of PAC Codes

“…Recently, Arikan proposed polarization-adjusted convolutional (PAC) codes [ 3 ], which used a convolutional pre-transformation before the polarization network, effectively alleviating the capacity loss caused by the slow polarization of the short and medium length channels. Although the sequential decoding algorithm used in this paper has high complexity, the performance of PAC (128, 64) code is almost identical to the PPV bound [ 4 ]. Using list decoding and sequential decoding, PAC codes always show better performance than polar codes with the same blocklength [ 5 ].…”