Improved Constructions of Reed-Solomon Codes with Optimal Repair Bandwidth

TL;DR

This paper introduces an improved construction for Reed-Solomon MSR codes that reduces subpacketization and expands feasible parameters, enhancing efficiency in distributed storage repair bandwidth.

Contribution

It eliminates the previous congruence condition on primes, significantly reducing subpacketization and broadening parameter options for RS-MSR codes.

Findings

Reduces subpacketization by a factor of (s)^n.

Broadens the range of feasible parameters for RS-MSR codes.

Maintains optimal repair bandwidth with improved construction.

Abstract

Maximum-distance-separable (MDS) codes are widely used in distributed storage, yet naive repair of a single erasure in an $[n,k]$ MDS code downloads the entire contents of $k$ nodes. Minimum Storage Regenerating (MSR) codes (Dimakis et al., 2010) minimize repair bandwidth by contacting $d>k$ helpers and downloading only a fraction of data from each. Guruswami and Wootters first proposed a linear repair scheme for Reed-Solomon (RS) codes, showing that they can be repaired with lower bandwidth than the naive approach. The existence of RS codes achieving the MSR point (RS-MSR codes) nevertheless remained open until the breakthrough construction of Tamo, Barg, and Ye, which yields RS-MSR codes with subpacketization $\ell = s \prod_{i=1}^n p_i$, where $p_i$ are distinct primes satisfying $p_i \equiv 1 \pmod{s}$ and $s=d+1-k$. In this paper, we present an improved construction of RS-MSR codes by eliminating the congruence condition $p_i \equiv 1 \pmod{s}$. Consequently, our construction reduces the subpacketization by a multiplicative factor of $\phi(s)^n$ ( $\phi(\cdot)$ is Euler's totient function) and broadens the range of feasible parameters for RS-MSR codes.