Recovery Set Structures and Service Rates of Codes Obtained by the Plotkin-type Construction

Abstract

In distributed storage systems, redundancy enables a single data object to be reconstructed using several disjoint groups of servers. The resulting service rate region (SRR) captures all combinations of object request rates that the system can support simultaneously without overloading any individual server. We analyze the SRR of binary codes obtained via the Plotkin-type construction. We demonstrate that each recovery set for an information object of the Plotkin-type code $\mathcal{C}_p$ induces a corresponding recovery set for the same data object characterized by the underlying code $\mathcal{C}$. Furthermore, by characterizing the recovery sets structure of $\mathcal{C}_p$ in terms of the recovery set structure of $\mathcal{C}$, we identify the additional recovery sets introduced by this construction. Using the associated recovery hypergraphs, we establish bounds on the SRRs of $\mathcal{C}_p$ and iterated code $\mathcal{C}_{p^m}$ in terms of the SRR of $\mathcal{C}$. We consider the family of first-order binary Reed-Muller codes and show how the recovery structure, and consequently, the service rates of R$(1,m)$ for arbitrary $m$ can be recursively derived from a generator matrix of R$(1,2)$ through our results for successive Plotkin-type constructions.