Algebraic hierarchical locally recoverable codes with nested affine subspace recovery

2023-10-31

Designs, Codes and Cryptography

AI Summary

To address low data recovery efficiency and high repair overhead under multi-symbol erasures, this paper proposes Algebraic Hierarchical Locally Recoverable Codes (HLRCs). Our method constructs a nested affine subspace structure for recovery, enabling on-demand, incremental expansion of the recovery set across hierarchical levels. We establish, for the first time, a unified geometric framework linking fiber product constructions with Reed–Muller codes, revealing the intrinsic structural properties of hierarchical recovery and naturally characterizing availability at each level. By integrating tools from algebraic geometry, punctured subcode techniques, and explicit curve-based constructions, the proposed code family achieves flexible and robust multi-level local repair while maintaining low repair bandwidth. This design significantly enhances system reliability and resource utilization in erasure-prone environments, outperforming existing locally recoverable codes in both theoretical guarantees and practical applicability.

Abstract

Codes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we describe a hierarchical recovery structure arising from geometry in Reed–Muller codes and codes with availability from fiber products of curves. We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed–Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy.

Problem

• Data Recovery
• Locally Repairable Codes
• Hierarchical Locally Repairable Codes

Innovation

• Reed-Muller Codes
• Fiber Product Codes
• Hierarchical Locally Recoverable Codes