Conjectured constant multiplicities within the punctured-Golay families

Prove that, for every m\ge2, the families D_4 and D_5 are respectively infinite families of (3,72)-APMCF and (2,24)-APMCF codes, and that, for every m\ge3, D_3 is an infinite family of (4,180)-APMCF codes.

Background

Computational results establish the claimed multiplicities for several initial lifted codes: \lambda_{9,2}=\lambda_{9,3}=72, \lambda_{8,2}=\lambda_{8,3}=24, and \lambda_{10,3}=180. The authors conjecture that these values persist throughout the corresponding infinite families D_4, D_5, and D_3, respectively, but do not prove the general assertions.

References

This gives rise to conjecture that for $m\ge2$, $D_4$ and $D_5$ are infinite families of $(3,72)$-APMCF and $(2,24)$-APMCF codes, respectively, and similarly, for $m\ge3$, $D_3$ is infinite family of $(4,180)$-APMCF codes.

— 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(ii), Section 5; reiterated as an open problem in Section 6