Efficient decoding for existing purity-testing code constructions

Determine whether the non-LDPC purity-testing code constructions based on rational curves and classical error-correcting codes admit efficient decoding algorithms.

Background

The paper discusses purity-testing code ensembles used in quantum message authentication. Earlier constructions by Barnum et al. use rational curves in projective space, while a later construction simplifies the approach using classical error-correcting codes. These constructions are not LDPC, and the paper identifies the efficiency of their decoding algorithms as unresolved; the paper’s own QLDPC construction provides a separate family with polynomial-time decoding.

References

However, neither of these constructions is LDPC, and it is not known if they admit efficient decoding algorithms.

— Random Quantum LDPC Codes Approaching the Gilbert-Varshamov Bound  (2610.02648 - Mittal et al., 2 Oct 2026) in Section “Purity Testing and Authentication,” subsection “Decoding”