Researchers have introduced a conditional density‑evolution Gaussian approximation (CDEGA) and a new dangerous‑path occupancy metric to analyze post‑decoding errors in PAC codes. CDEGA extends the standard density‑evolution Gaussian approximation to conditions where decoding errors already exist, allowing the study of how information bits fail during successive‑cancellation‑list (SCL) decoding. The dangerous‑path metric counts how often an information bit is incorrectly decoded on paths that survive to the final list, giving a statistic that is directly tied to finite‑list performance.

Using these tools, the authors built two search procedures. A CDEGA‑based information‑set search iteratively swaps bits between the information and frozen sets to maximise a minimum expected path‑metric over all error states. A separate safe‑precoder construction adds sparse, time‑varying convolutional edges guided by the dangerous‑path statistics. Together, the methods can be run iteratively for a modest number of steps or until the frame‑error‑rate (FER) no longer improves.

When applied to codes of length N ≤ 512 across a range of rates and list sizes, the new designs yield at least a 0.5 dB improvement at FER = 10⁻³ over baseline PAC codes that use the standard DEGA. For a specific 64‑bit, 32‑information‑bit, list‑size‑32 code, the best design (Algorithm 2 with a sparse precoder) lifts the FER curve by roughly 0.75 dB at the same target FER.

The gains are most pronounced at moderate list sizes (≈32). At very large lists where the decoder approaches maximum likelihood performance, additional benefits diminish, and the weight spectra of the improved codes remain superior to standard PAC codes. At very high design SNRs, the Monte‑Carlo simulation used to estimate danger‑path counts suffers from high variance because errors become rarer, limiting the accuracy of the approach.