Improved Constructions of Reed-Solomon Codes with Optimal Repair Bandwidth

TL;DR

The paper addresses scalar Reed-Solomon MSR codes and the subpacketization bottleneck by removing the TYB17 requirement that primes satisfy $p_i\equiv1\pmod{s}$. It introduces a basis-transformation framework built from Euclidean Square Partition, Reshape, and Interference to construct an $\mathbb{F}$-subspace $S$ of dimension $p$ in a suitable extension, enabling RS-MSR codes with subpacketization $\ell = s\prod_{i=1}^n p_i$ for primes $p_i>s$ without the congruence constraint. The key contributions are the generalized subspace construction and the explicit transformation toolkit, which reduce the TYB17 subpacketization by a factor of $\phi(s)^n$ and widen the feasible parameter range. This advances MSR capabilities for RS codes in distributed storage by lowering storage and repair overhead while maintaining the MDS property.

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 $φ(s)^n$ ( $φ(\cdot)$ is Euler's totient function) and broadens the range of feasible parameters for RS-MSR codes.

Figures (3)

• Figure 1: Original ($p \times s$)
• Figure 2: Target ($s \times p$)
• Figure 3: The transformation procedure for $(p,s)=(7,5)$.

Theorems & Definitions (23)

• Theorem 1: Guruswami16
• Theorem 2: TYB17
• Lemma 1: Lidl94
• Lemma 2: TYB17
• Remark 1
• Lemma 3
• Definition 1: Euclidean Square Partition
• Definition 2: Reshape
• Definition 3
• Example 1