Optimal exceptional cliques in higher-dimensional integer matrix rings

Determine whether the integer matrix ring Mat_s(Z) contains an exceptional clique of size 2^s for every integer s greater than 4.

Background

The general upper bound for exceptional cliques in Mat_s(Z) is 2s. The paper establishes existence of cliques attaining this bound for s=1, 2, 3, and 4, and explains that such optimal cliques would improve applications to zero-knowledge proofs and secret sharing.

The authors explicitly state that the existence question remains unresolved for every dimension beyond 4.

References

As shown in this work, also for $s=4$, $\textup{Mat}_s()$ has an exceptional clique of size $2s$, while the question is open for all other $s>4$.

— On exceptional cliques in matrix rings  (2608.24586 - Boutros et al., 25 Aug 2026) in Section 9 (Exceptional cliques in cryptography)