A Path Metric Based Construction of Polarization-Adjusted Convolutional Codes

We propose an approach to understand and exploit Polarization-Adjusted Convolutional (PAC) Codes that is directly tied to decoders with memory, specifically Successive Cancellation List (SCL) decoding. The crux of our approach is to use a modified Density Evolution Gaussian Approximation (DEGA) to account for errors in the decoding process and accurately track the path metrics (PMs) likely to incur decoding errors. Our approach not only explains the benefits provided by the use of the rate one precoding, but also provides new insight into why certain information sets perform well under PAC. We leverage the approach to design new information sets, and in particular, we design a
$($
128,42,
$L=8)$
code that offers half a
$\text{dB}$
gain over the state-of-the-art at a Frame Error Rate (FER) of
$\overline{1}0^{-3}$
and outperforms the Reed-Muller (RM) set with
$L=32$
. We also draw connections between our approach and works studying the minimum weight of PAC codes.