Calculating the I/O Cost of Linear Repair Schemes for RS Codes Evaluated on Subspaces via Exponential Sums

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 ${\mathbb{F}}_{q^{\mathrm{\ell }}}$ with $r=2$ or $3$ parities. These bounds are exactly matched in the cases $r=2,\mathrm{\ell }-d+1\mid \mathrm{\ell }$ and $r=3,d=\mathrm{\ell }$ or $\mathrm{\ell }-d+2\mid \mathrm{\ell }$, 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.