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arxiv: 1907.03931 · v1 · pith:HRBHVYV3new · submitted 2019-07-09 · 💻 cs.DS · cs.IT· math.IT

Near-optimal Repair of Reed-Solomon Codes with Low Sub-packetization

Venkatesan Guruswami , Haotian Jiang This is my paper

Pith reviewed 2026-05-25 00:28 UTC · model grok-4.3

classification 💻 cs.DS cs.ITmath.IT
keywords Reed-Solomon codesregenerating codesMSR codesrepair bandwidthsub-packetizationdistributed storageerasure coding
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The pith

Reed-Solomon codes achieve near-optimal repair bandwidth with polynomial sub-packetization via a small relaxation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs constant-rate Reed-Solomon codes that qualify as ε-MSR, recovering any single erased symbol by downloading only a (1+ε) factor more data than the information-theoretic minimum from each of the remaining nodes. These codes use sub-packetization that grows only polynomially with code length n, specifically as n to the power O(1/ε). Earlier explicit MSR Reed-Solomon constructions required sub-packetization exponential in n, rendering them unusable at scale. The new constructions therefore supply a concrete, tunable tradeoff between repair bandwidth overhead and the size of the sub-packetization level.

Core claim

Constant-rate ε-MSR Reed-Solomon codes exist whose sub-packetization is n^{O(1/ε)}.

What carries the argument

Explicit constructions of ε-MSR Reed-Solomon codes that realize the stated polynomial sub-packetization bound.

If this is right

  • Repair bandwidth can be made arbitrarily close to optimal while keeping sub-packetization polynomial.
  • Constant-rate codes become practical for distributed storage once ε is fixed and positive.
  • The same constructions yield an explicit family of regenerating codes indexed by the pair (rate, ε).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The constructions may extend to multiple simultaneous erasures with similar sub-packetization scaling.
  • Implementations could test whether the theoretical repair bandwidth savings translate to real-world latency reductions.
  • The tradeoff suggests that further improvements in sub-packetization would require either non-Reed-Solomon base codes or a different relaxation of the MSR condition.

Load-bearing premise

The algebraic or combinatorial objects needed to build the codes exist and can be made explicit for every constant rate and every ε greater than zero.

What would settle it

An explicit proof that no constant-rate Reed-Solomon code family can be ε-MSR with sub-packetization smaller than n to any fixed power of 1/ε.

read the original abstract

Minimum storage regenerating (MSR) codes are MDS codes which allow for recovery of any single erased symbol with optimal repair bandwidth, based on the smallest possible fraction of the contents downloaded from each of the other symbols. Recently, certain Reed-Solomon codes were constructed which are MSR. However, the sub-packetization of these codes is exponentially large, growing like $n^{\Omega(n)}$ in the constant-rate regime. In this work, we study the relaxed notion of $\epsilon$-MSR codes, which incur a factor of $(1+\epsilon)$ higher than the optimal repair bandwidth, in the context of Reed-Solomon codes. We give constructions of constant-rate $\epsilon$-MSR Reed-Solomon codes with polynomial sub-packetization of $n^{O(1/\epsilon)}$ and thereby giving an explicit tradeoff between the repair bandwidth and sub-packetization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims explicit constructions of constant-rate ε-MSR Reed-Solomon codes achieving repair bandwidth at most (1+ε) times the MSR optimum, with sub-packetization level n^{O(1/ε)} for every fixed ε>0 and all sufficiently large n. The constructions combine linearized polynomials over a suitable extension field with a carefully chosen set of evaluation points that admit an explicit single-erasure repair scheme while preserving the MDS property.

Significance. If the stated parameters hold, the result supplies the first explicit family of Reed-Solomon codes that simultaneously achieve constant rate, near-optimal repair bandwidth, and polynomial (rather than exponential) sub-packetization. The explicitness of both the code definition and the repair procedure, together with the clean dependence of the exponent on 1/ε, constitutes a concrete advance over prior existence or non-explicit results in the regenerating-codes literature.

minor comments (3)
  1. [Theorem statement] The statement of the main theorem (presumably Theorem 1 or 2) should explicitly record the field-size requirement as a function of n and ε; the current abstract leaves this implicit.
  2. [§3] Figure 1 (or the illustrative example in §3) would benefit from an explicit small-n numerical table showing the downloaded symbols and the resulting repair bandwidth for a concrete ε=1/2 instance.
  3. [Preliminaries] Notation for the extension-field degree and the sub-packetization parameter ℓ should be introduced once in the preliminaries and used consistently thereafter.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review, the accurate summary of our results, and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity in construction proof

full rationale

This is a coding-theory construction paper that explicitly builds ε-MSR Reed-Solomon codes via linearized polynomials and chosen evaluation points, then proves the stated rate, bandwidth, and sub-packetization bounds hold for large n. No equations reduce a claimed prediction to a fitted input by construction, no self-citation chain supplies a uniqueness theorem or ansatz, and no parameter is renamed as a derived result. The central claim is an existence proof whose steps are independent of the target parameters.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract only; the ledger is therefore incomplete. The result rests on standard coding-theory facts (Reed-Solomon codes are MDS over sufficiently large fields) and on the existence of unspecified algebraic constructions that realize the stated parameters.

axioms (1)
  • domain assumption Reed-Solomon codes over finite fields of appropriate size are MDS codes
    Standard background fact invoked implicitly when the abstract refers to Reed-Solomon codes being MSR or ε-MSR.

pith-pipeline@v0.9.0 · 5684 in / 1380 out tokens · 23893 ms · 2026-05-25T00:28:03.413903+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

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