New Centralized MSR Codes With Small Sub-packetization
Yaqian Zhang
TL;DR
The work tackles efficient centralized repair for MSR codes in distributed storage by constructing explicit MDS array codes with small sub-packetization across broad parameter regimes. It develops a progression of codes, from C1 supporting multiple repair degrees to C2 and C3 for general and all h,d, and culminates in a universal scheme C4 that achieves (h,d)-optimal repair for every feasible pair, all while keeping sub-packetization modest. Central to the methods are parity-check structures that induce generalized Reed-Solomon code properties and organized repair steps, including grouping and Hadamard-based refinements for special cases. Collectively, the results improve sub-packetization bounds over prior work and expand the practical applicability of centralized MSR repair in large-scale storage systems.
Abstract
Centralized repair refers to repairing $h\geq 2$ node failures using $d$ helper nodes in a centralized way, where the repair bandwidth is counted by the total amount of data downloaded from the helper nodes. A centralized MSR code is an MDS array code with $(h,d)$-optimal repair for some $h$ and $d$. In this paper, we present several classes of centralized MSR codes with small sub-packetization. At first, we construct an alternative MSR code with $(1,d_i)$-optimal repair for multiple repair degrees $d_i$ simultaneously. Based on the code structure, we are able to construct a centralized MSR code with $(h_i,d_i)$-optimal repair property for all possible $(h_i,d_i)$ with $h_i\mid (d_i-k)$ simultaneously. The sub-packetization is no more than ${\rm lcm}(1,2,\ldots,n-k)(n-k)^n$, which is much smaller than a previous work given by Ye and Barg ($({\rm lcm}(1,2,\ldots,n-k))^n$). Moreover, for general parameters $2\leq h\leq n-k$ and $k\leq d\leq n-h$, we further give a centralized MSR code enabling $(h,d)$-optimal repair with sub-packetization smaller than all previous works.