Optimal Conversion Bandwidth for MDS Convertible Codes in the Split Regime with $r^F<k^F<r^I$

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 $[n^I,k^I]$ 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.