Optimal Conversion Bandwidth for MDS Convertible Codes in the Split Regime with rF<kF<rI
Abstract
Erasure codes are widely used in distributed storage systems to provide fault tolerance. An [n,k] erasure code encodes k data symbols into n coded symbols and distributes them across n storage nodes. Once the code parameters are fixed, the achievable fault tolerance is also fixed. However, the failure rates of storage nodes may vary over time, and dynamically adapting the code parameters to these variations can substantially reduce storage overhead. Motivated by this observation, Maturana and Rashmi introduced convertible codes, which allow an [nI,kI] initial code with redundancy rI=nI−kI to be transformed into an [nF,kF] final code with redundancy rF=nF−kF, while preserving the required code properties. Convertible codes whose initial and final codes are both MDS codes are called MDS convertible codes. They are of particular interest because MDS codes provide the maximum erasure tolerance for a given amount of storage overhead. Several works have established lower bounds and constructions for the bandwidth cost of MDS convertible codes in the split regime. However, the tight bound in the parameter range rF<kF<rI remains unknown. In this work, we establish a family of new rank inequalities for stable MDS convertible codes with linear conversion procedures and derive an improved lower bound on the bandwidth cost consisting of three cases for this remaining range. We prove that the bound is tight by presenting three explicit constructions, one for each case.
Related
- Entropy/Length Profile Bounds for the Conversion of (Non)linear Codes in the Merge Regime
- Constructions of Secure Convertible Codes
- Exact Storage-Repair-Bandwidth Tradeoffs for (6,5,5) and (7,6,6) Exact-Repair Regenerating Codes
- From Linear Exact Repair to Mixed-Alphabet Codes: The Bandwidth-Subpacketization Tradeoff of MDS Array Codes
- The circular law for non-Hermitian random band matrices: optimal bandwidth, periodic profile and discrete law
BibTeX
@misc{wang2026optimal,
title = {{Optimal Conversion Bandwidth for MDS Convertible Codes in the Split Regime with $r^F<k^F<r^I$}},
author = {Lewen Wang and Sihuang Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08349},
eprint = {2610.08349},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08349},
}
RIS
TY - UNPB
TI - Optimal Conversion Bandwidth for MDS Convertible Codes in the Split Regime with r^F<k^F<r^I
AU - Lewen Wang
AU - Sihuang Hu
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08349
UR - https://arxiv.org/abs/2610.08349
AB - Erasure codes are widely used in distributed storage systems to provide fault tolerance. An [n,k] erasure code encodes k data symbols into n coded symbols and distributes them across n storage nodes. Once the code parameters are fixed, the achievable fault tolerance is also fixed. However, the failure rates of storage nodes may vary over time, and dynamically adapting the code parameters to these variations can substantially reduce storage overhead. Motivated by this observation, Maturana and Rashmi introduced convertible codes, which allow an [n^I,k^I] initial code with redundancy r^I=n^I−k^I to be transformed into an [n^F,k^F] final code with redundancy r^F=n^F−k^F, while preserving the required code properties. Convertible codes whose initial and final codes are both MDS codes are called MDS convertible codes. They are of particular interest because MDS codes provide the maximum erasure tolerance for a given amount of storage overhead. Several works have established lower bounds and constructions for the bandwidth cost of MDS convertible codes in the split regime. However, the tight bound in the parameter range r^F<k^F<r^I remains unknown. In this work, we establish a family of new rank inequalities for stable MDS convertible codes with linear conversion procedures and derive an improved lower bound on the bandwidth cost consisting of three cases for this remaining range. We prove that the bound is tight by presenting three explicit constructions, one for each case.
ER -