Closed-form enumeration of higher-weight polar-code codewords

Derive closed-form expressions for the number of codewords of weight 2 and higher across all polar-code rates.

Background

The tutorial surveys closed-form weight-enumeration results for minimum-weight codewords and for selected low-weight ranges, primarily in low- and high-rate polar codes where the full weight distribution can sometimes be obtained. It identifies the extension to codewords of weight 2 and beyond, uniformly over all code rates, as an unresolved combinatorial problem.

References

However, deriving closed-form expressions for $2$-weight codewords {and beyond} across all code rates remains an open problem.

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