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

2026-10-01

AI Summary

This study addresses the unresolved boundaries of the service rate region and the structure of recovery sets for Plotkin-constructed codes in distributed storage systems. By integrating coding theory with hypergraph models, the proposed approach correlates the recovery sets of constituent codes with those of the constructed code, revealing the additional recovery set structures introduced by the Plotkin construction. This framework establishes both upper and lower bounds on the service rate region and enables the recursive derivation of performance metrics for Reed-Muller codes. The primary contributions include the formal characterization of theoretical bounds for the service rate region of constructed codes and the development of an efficient recursive analysis methodology. Ultimately, this work provides a novel analytical framework for evaluating and optimizing the performance of higher-order codes in distributed storage architectures.

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.

Problem

• Distributed storage systems
• Service rate region
• Recovery sets
• Plotkin-type construction

Innovation

• Plotkin-type construction
• Service rate region
• Recovery sets
• Recovery hypergraphs
• Reed-Muller codes