Attainment or asymptotic attainment of the generalized graph-code distance bound

Determine whether there exists a generalized graph code on an ℓ-partite graph with ℓ ≥ 3 whose minimum distance attains the lower bound in Theorem 1.1, or, if no such example exists, whether the lower bound is attained asymptotically.

Background

Theorem 1.1 establishes a lower bound for the minimum distance of a generalized graph code constructed from an ℓ-partite graph and identical component codes. The bound depends on the component-code minimum distance, the graph size and valency, and the second-largest adjacency eigenvalue.

The paper states that no example attaining this lower bound had been obtained. Problem 5.1 asks whether exact attainment occurs for ℓ ≥ 3 and, failing that, whether a sequence of constructions can approach the bound asymptotically.