Exact Storage-Repair-Bandwidth Tradeoffs for $(6,5,5)$ and $(7,6,6)$ Exact-Repair Regenerating Codes

This note provides exact information-theoretic outer bounds that completely characterize the storage-repair-bandwidth tradeoff regions for the cases $(n,k,d)=(6,5,5)$ and $(7,6,6)$. The layered code proposed by Tian et al. achieves the tradeoff and is thus optimal. GPT-6 Astra assisted in the discovery of these bounds, based on a new conditioning argument and a structured search over the restricted subsets of entropy linear program constraints. The proofs are given as exact integer-valued certificates, which are difficult for direct human verification, and a Python verifier is included as part of the package, together with the certificates.