Classification of perfect Lee codes in higher dimensions

Prove or disprove the conjecture that, for every prime p=2d+1 with p≥13, the algebraically constructed radius-one perfect Lee codes in Z^d exhaust all perfect configurations of the low-activity nearest-neighbor maximal hard-core model.

Background

In the low-activity nearest-neighbor maximal hard-core model on Zd, perfect configurations correspond to radius-one perfect Lee codes, or tilings of Zd by crosses. For prime 2d+1, the paper describes a standard algebraic construction of such configurations and notes that it gives all perfect configurations in dimensions d=2, 3, and 5 based on known results. The authors then state a conjecture for all larger prime values.

References

For $p\ge 13$ prime, it is conjectured (seeConjecture~3) that this construction exhausts all perfect configurations.

— A general framework for crystallization in maximal hard-core models  (2610.11042 - Barg et al., 8 Oct 2026) in Section 5, Examples, subsection “The hypercubic lattices Z^d at low activity”