Exact universal replica threshold in the higher-multiplicity case

Determine the exact universal replica threshold q^*(A,p) for a fixed zero-diagonal representative A of an SBIBD(v,k,lambda) when p>=3, lambda>1, and A contains no pairwise separated set of size p, thereby resolving the remaining gap between the established lower and upper bounds.

Background

The paper defines q*(A,p) as the smallest number of replicas per object that guarantees recovery from every legal placement and every failure set of at most p nodes for a fixed zero-diagonal SBIBD representative A. For p>=3, the authors classify the threshold using pairwise separated sets and private target permutations.

When A contains no pairwise separated set of size p and lambda>1, the paper establishes only the bounds max{p, lambda+1} <= q*(A,p) <= lambda(p-1)+1. The authors explain that the unresolved value depends on finer local incidence information involving higher-order intersections and the locations of failed parity hosts, which is not captured by the structures analyzed in the paper.

References

The exact threshold for a fixed representative remains open when $p\ge3$, $\lambda>1$, and $A$ contains no pairwise separated set of size~$p$. Appendix~\ref{app:p2-example} gives an explicit zero-diagonal $\SBIBD(7,4,2)$ representative that realizes the case with a reciprocal pair for $p=2$.

— Replica Thresholds for Stripeless Erasure Coding Based on Symmetric Block Designs  (2609.29149 - Luo et al., 24 Sep 2026) in Section 3.4, subsection “Summary of thresholds for fixed representatives”; Section 6, Conclusion