MDS Generalized Convertible Code

TL;DR

This work generalizes convertible codes by allowing initial and final MDS codes with different parameters, enabling rate adaptation to device failure rates. It derives new lower bounds on access cost in both merge and split regimes and provides a parity-check-matrix based necessary-and-sufficient characterization of access-optimal merge-convertible codes. An explicit construction using extended generalized Reed-Solomon codes yields access-optimal MDS generalized merge-convertible codes with optimal field size, broadening parameter regimes beyond prior results. The results have practical implications for dynamic redundancy in distributed storage, offering design principles and concrete constructions to minimize I/O during code conversion while preserving MDS properties.

Abstract

In this paper, we consider the convertible codes with the maximum distance separable (MDS) property, which can adjust the code rate according to the failure rates of devices. We first extend the notion of convertible codes to allow initial and final codes with different parameters. Then, we investigate the relationship between these parameters and thus establish new lower bounds on the access cost in the merge and split regimes. To gain a deeper understanding of access-optimal MDS convertible codes in the merge regime, we characterize them from the perspective of parity check matrices. Consequently, we present a necessary and sufficient condition for the access-optimal MDS convertible code in the merge regime. Finally, as an application of our characterization, we construct MDS convertible codes in the merge regime with optimal access cost based on the extended generalized Reed-Solomon codes.

Figures (3)

• Figure 1: A $(2,2)_q$ convertible code for Example 1. Each block is a physically independent device and stores a single symbol. The red devices store unchanged symbols. The blue devices store read symbols. The green devices store written symbols. Notice that the symbols in yellow devices are overwritten by new symbols or released after the conversion procedure, which are called retired symbols in Maturana2020a. Moreover, there is an idle device, that is released symbol after conversion procedure, which is exactly the storage overhead saved.
• Figure 2: The relationship among code symbols used in the proof of Theorem \ref{['thm: merge-convertible: lower bound']} for $i=1$, where $\mathcal{F}_j\subseteq\{j\}\times[n_{I_j}]$ such that $|\mathcal{F}_j|=k_{I_j}, \mathcal{U}_{j,1}\subseteq\mathcal{F}_j$ for $j\in[t_1]\setminus\{1\}$, and $\mathcal{G}_1=\mathcal{U}_{1,1}\cap\mathcal{R}_{1,1}$, and $\mathcal{N}_1=\bigcup_{j\in[t_1]\setminus\{1\}}\mathcal{U}_{j,1}$.
• Figure 3: The relationship among code symbols used in the proof of Theorem \ref{['thm 1']}, where $\alpha\in\mathcal{U}_{1,j}\setminus(\bigcup_{j\in[t_2]}\mathcal{R}_{1,j})$, and $\mathcal{F}\subseteq\mathcal{U}_{1,j}\cup\mathcal{W}_j$ such that $|\mathcal{F}|\ge k_{F_j}, \mathcal{W}_j\subseteq\mathcal{F}, \alpha\not\in\mathcal{F}$, and $\mathcal{G}=\mathcal{U}_{1,j}\cap(\bigcup_{j\in[t_2]}\mathcal{R}_{1,j})$, and $\mathcal{H}\subseteq\mathcal{U}_{1,j}\setminus\mathcal{G}$ such that $|\mathcal{H}|=r_{I_1}+1$.

Theorems & Definitions (27)

• Lemma 1: Huffman2003
• Definition 1
• Definition 2
• Lemma 2: Huffman2003
• Definition 3: Generalized Convertible Code
• Definition 4
• Remark 1
• Definition 5: Stable Generalized Convertible Code
• Definition 6: Access Cost
• Example 1