Closed-form enumeration for pre-transformed polar codes

Develop closed-form formulas for enumerating the weight distributions of pre-transformed polar codes, including CRC-polar codes and PAC codes.

Background

The existing closed-form enumeration formulas discussed in the tutorial apply to plain polar codes, whose algebraic structure supports orbit-based analysis. The paper explicitly notes that extending such enumeration to pre-transformed constructions, specifically CRC-polar and PAC codes, has not yet been addressed.

References

Moreover, these existing formulae apply only to plain polar codes, which possess elegant algebraic structure. The {closed-form} enumeration of pre-transformed polar codes - such as CRC-polar codes and PAC codes, remains unaddressed.

— A Tutorial on Weight Structure of Polar Codes  (2608.18493 - Rowshan et al., 19 Aug 2026) in Section: Outlook and Future Directions