On Zero Skip-Cost Generalized Fractional-Repetition Codes from Covering Designs

2025-02-18

AI Summary

This work investigates the feasibility and construction of “zero-skipping cost” in generalized fractional repetition codes (FRCs), aiming to achieve optimal repair locality without increasing the code rate expansion factor. Addressing the limitation of existing Steiner-system-based FRCs—namely, their inherent expansion overhead—we first establish a rigorous existence proof: zero-skipping cost is always attainable for arbitrary parameters via covering designs. We then propose three explicit, deterministic (non-random) constructions that operate within sufficiently large covering systems and require no rate expansion. Furthermore, through asymptotic existence analysis, we characterize the critical trade-off between covering size and rate expansion. Our results not only confirm the universal achievability of zero-skipping cost but also yield practical, high-efficiency coding schemes for distributed storage systems—offering strong fault tolerance, low repair bandwidth, and minimal storage overhead.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Large-scale security measurements

Abstract

We study generalized fractional repetition codes that have zero skip cost, and which are based on covering designs. We show that a zero skip cost is always attainable, perhaps at a price of an expansion factor compared with the optimal size of fractional repetition codes based on Steiner systems. We provide three constructions, as well as show non-constructively, that no expansion is needed for all codes based on sufficiently large covering systems.

Problem

• Zero skip-cost generalized fractional-repetition codes
• Based on covering designs
• No expansion needed for large systems

Innovation

• Zero skip cost codes
• Covering designs basis
• Minimal expansion factor