Linear Code Conversion in the Merge Regime: General Bounds and Reed--Muller Constructions

2026-06-25

AI Summary

This work addresses the high recoding overhead incurred during erasure code parameter adjustment in distributed storage systems by investigating scalar linear code conversion for arbitrary linear codes under merging scenarios. By introducing generalized Hamming weights to characterize the growth of subcode support sets, the study establishes universal lower bounds on read and write overheads, overcoming the limitations of traditional analyses that rely solely on minimum distance. Within this framework, the authors not only recover known bounds for special cases but also derive tighter bounds when the target code exhibits nontrivial jumps in generalized Hamming weights. Furthermore, by leveraging Plotkin decomposition and the structure of Reed–Muller codes, they explicitly construct convertible codes that achieve the write overhead lower bound under natural parameters, while also attaining optimal single-block read overhead.

Abstract

Erasure codes are a core component of most existing large-scale distributed storage systems, ensuring reliability against node failures. Recent work has shown that adapting code parameters to changing node failure rates can lead to significant storage savings. The default approach is to re-encode the data under a new code, which consumes substantial system resources. Code conversion was introduced to reduce this cost. However, existing work has mainly focused on conversions within specific classes of codes. In this paper, we study scalar linear code conversion in the merge regime for arbitrary linear codes. We derive universal lower bounds on the write and read costs in terms of unchanged and read symbols. The bounds are refined using generalized Hamming weights, which capture support-growth properties of subcodes and can give sharper estimates than minimum-distance-only arguments. We show that the framework recovers known bounds for important special cases and can be strictly stronger when the final code has nontrivial jumps in its generalized Hamming weight hierarchy. We then apply the framework to Reed-Muller codes and construct explicit Reed-Muller convertible codes using the Plotkin decomposition. For a natural Reed-Muller parameter regime, the construction attains the derived write-cost lower bound. For the read cost, the generalized-Hamming-weight analysis is sharp for one initial block, while a gap remains for the other block.

Problem

• code conversion
• linear codes
• erasure codes
• generalized Hamming weights
• Reed-Muller codes

Innovation

• linear code conversion
• generalized Hamming weights
• Reed–Muller codes
• merge regime
• storage efficiency