Set Transformation: Trade-off Between Repair Bandwidth and Sub-packetization

TL;DR

This work introduces set transformation as a general method to convert any MDS code into a set transformed MDS (ST-RS) code with flexible sub-packetization $α$ and improved single-node repair efficiency. By partitioning an $α \times n$ array into sub-arrays, applying set pairwise combinations, and storing $α$ symbols per node, the resulting ST-RS$(n,k,α)$ codes preserve the MDS property while achieving lower repair bandwidth than prior elastic and related constructions. The authors derive repair procedures and lower bounds for ST-RS, establish a field-size condition to guarantee MDS, and show through comparisons that ST-RS attains smaller field sizes and reduced bandwidth than competing schemes across representative parameters. These results offer a practical pathway to high-rate, low-bandwidth MDS array codes suitable for scalable storage systems, with potential for recursion to reach even finer sub-packetization levels.

Abstract

Maximum distance separable (MDS) codes facilitate the achievement of elevated levels of fault tolerance in storage systems while incurring minimal redundancy overhead. Reed-Solomon (RS) codes are typical MDS codes with the sub-packetization level being one, however, they require large repair bandwidth defined as the total amount of symbols downloaded from other surviving nodes during single-node failure/repair. In this paper, we present the {\em set transformation}, which can transform any MDS code into set transformed code such that (i) the sub-packetization level is flexible and ranges from 2 to $(n-k)^{\lfloor\frac{n}{n-k}\rfloor}$ in which $n$ is the number of nodes and $k$ is the number of data nodes, (ii) the new code is MDS code, (iii) the new code has lower repair bandwidth for any single-node failure. We show that our set transformed codes have both lower repair bandwidth and lower field size than the existing related MDS array codes, such as elastic transformed codes \cite{10228984}. Specifically, our set transformed codes have $2\%-6.6\%$ repair bandwidth reduction compared with elastic transformed codes \cite{10228984} for the evaluated typical parameters.

Figures (3)

• Figure 1: Sub-array allocation for $\alpha \times \beta$ array with $\alpha\leq \beta<2\alpha$.
• Figure 2: Set Transformation on $4\times 6$ array.
• Figure 3: ST-RS$(14, 10,3)$ code with two square sub-arrays and two non-square sub-arrays.

Theorems & Definitions (12)

• Example 1
• Example 2
• Theorem 3
• proof
• Theorem 4
• proof
• Lemma 5
• proof
• Theorem 6
• proof