Constant multiplicities for the D_3, D_4, and D_5 families

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

Background

The authors compute the APMCF multiplicities for selected lifted punctured Golay codes: 72 for G_{9,2} and G_{9,3}, 24 for G_{8,2} and G_{8,3}, and 180 for G_{10,3}. They observe that these values suggest constant multiplicities throughout the corresponding infinite families, but the available computations do not establish the claim for every lift. The resulting conjecture is explicitly listed among the paper’s open problems.

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