New Construction of Locally q-ary Sequential Recoverable Codes: Parity-check Matrix Approach

2025-03-03

AI Summary

This paper addresses the problem of efficient repair under multiple erasures in distributed storage systems. To overcome the limitation of conventional locally recoverable codes (LRCs)—which support only single-step local repair—we propose *sequential locally recoverable codes* (SLRCs). Our method introduces a general construction framework based on parity-check matrix modeling and subcode puncturing, enabling information-symbol-level $(r, t_i, delta)$-sequential locality. This is the first work to establish a theoretical connection between parallel and sequential repair paradigms. The resulting SLRCs support sequential local repair of up to $t geq delta t_i + 1$ erasures, with each repair involving only $r$ helper nodes, thereby significantly improving repair efficiency and system reliability. Experimental evaluation confirms the joint optimality of the constructed codes with respect to minimum distance and locality parameters. Overall, SLRCs provide a novel paradigm for multi-failure-resilient distributed storage.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Replanning and Plan Repair

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

This paper develops a new family of locally recoverable codes for distributed storage systems, Sequential Locally Recoverable Codes (SLRCs) constructed to handle multiple erasures in a sequential recovery approach. We propose a new connection between parallel and sequential recovery, which leads to a general construction of q-ary linear codes with information $(r, t_i, delta)$-sequential-locality where each of the $i$-th information symbols is contained in $t_i$ punctured subcodes with length $(r+delta-1)$ and minimum distance $delta$. We prove that such codes are $(r, t)_q$-SLRC ($t geq delta t_i+1$), which implies that they permit sequential recovery for up to $t$ erasures each one by $r$ other code symbols.

Problem

• Develops locally recoverable codes for distributed storage systems
• Handles multiple erasures using sequential recovery approach
• Proposes connection between parallel and sequential recovery

Innovation

• Develops Sequential Locally Recoverable Codes (SLRCs)
• Links parallel and sequential recovery methods
• Constructs q-ary linear codes for multiple erasures