Extendibility of regular hypercuboids

Determine for which parameters d, n, k, and δ every δ-regular hypercuboid H ∈ LHC_d(n,k) is extendible.

Background

A hypercuboid is δ-regular when the intersections of unused-symbol sets associated with axis-neighbouring lines have approximately the expected size. The authors suggest pseudorandomness as a possible mechanism ruling out the structured obstructions used in known nonextendibility constructions.

References

Question 13. For which d, n, k,δ does it follow that everyδ -regular H ∈ LHCd (n, k)must be extendible?

— Extendibility of Latin Hypercuboids  (2502.08868 - Bowtell et al., 13 Feb 2025) in Question 13, Section 5.3, p. 10