Calculating the I/O Cost of Linear Repair Schemes for RS Codes Evaluated on Subspaces via Exponential Sums
Zhongyan Liu, Jingke Xu, Zhifang Zhang
TL;DR
The paper addresses the I/O cost in repairing Reed-Solomon codes by linking it to exponential-sum evaluations of linear spaces, enabling a concise, computable I/O-cost formula for RS codes evaluated on subspaces. It introduces an $(m,t)$-normalized polynomial framework that standardizes repair polynomials and expresses $\gamma_{I/O}$ via additive-character sums, allowing Weil-bound-based lower bounds for two- and three-parity RS codes. The authors derive tight bounds in key regimes, characterize the optimal repair bandwidth for I/O-optimal schemes, and present constructions that beat prior bandwidth in certain full-length three-parity cases. Additionally, they construct I/O-efficient repairs for RS codes on $B$-linear subspaces, achieving the established lower bounds in several divisibility scenarios. The results advance understanding of the tradeoffs between I/O cost and repair bandwidth and enable more efficient distributed storage repair for RS codes in subspace settings.
Abstract
The I/O cost, defined as the amount of data accessed at helper nodes during the repair process, is a crucial metric for repair efficiency of Reed-Solomon (RS) codes. Recently, a formula that relates the I/O cost to the Hamming weight of some linear spaces was proposed in [Liu\&Zhang-TCOM2024]. In this work, we introduce an effective method for calculating the Hamming weight of such linear spaces using exponential sums. With this method, we derive lower bounds on the I/O cost for RS codes evaluated on a $d$-dimensional subspace of $\mathbb{F}_{q^\ell}$ with $r=2$ or $3$ parities. These bounds are exactly matched in the cases $r=2,\ell-d+1\mid\ell$ and $r=3,d=\ell$ or $\ell-d+2\mid\ell$, via the repair schemes designed in this work. We refer to schemes that achieve the lower bound as I/O-optimal repair schemes. Additionally, we characterize the optimal repair bandwidth of I/O-optimal repair schemes for full-length RS codes with two parities, and build an I/O-optimal repair scheme for full-length RS codes with three parities, achieving lower repair bandwidth than previous schemes.