Equality of APMCF multiplicities for lifted codes and their duals

Prove that, for the lifted Golay-related codes G_{n,m} and their duals G_{n,m}^\bot, the multiplicities \lambda_{n,m} and \lambda_{n,m}^\bot are equal.

Background

The paper observes matching multiplicities for several computed pairs of lifted codes and their duals, including the values 72, 24, and 180 in the cases listed in Remark 5.2(iii). On this basis, it formulates a conjecture that the APMCF multiplicities of each code and its dual always coincide, analogous to the equality of the numbers of minimum-weight codewords for an NMDS code and its dual.

References

This gives rise to the conjecture that for codes $G_{n,m}$ and $G_{n,m}\bot$ the values of $\lambda_{n,m}$ and $\lambda_{n,m}\bot$ are equal to each other; similarly to the number of codewords of minimal weight for an NMDS code and its dual, see Theorem~\ref{th25:NMDS}(iv).

— New infinite families of uniformly packed near-MDS codes and multiple coverings, based on the ternary Golay code  (2502.10223 - Davydov et al., 14 Feb 2025) in Remark 5.2(iii), Section 5; reiterated as an open problem in Section 6