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arxiv: 2604.15282 · v1 · submitted 2026-04-16 · 💻 cs.IT · math.IT

Bandwidth Cost of Locally Repairable Convertible Codes in the Global Merge Regime

Saransh Chopra , Shubhransh Singhvi , K.V. Rashmi This is my paper

Pith reviewed 2026-05-10 09:43 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords locally repairable codesconvertible codesbandwidth costglobal merge regimedistributed storageinformation-theoretic boundscode conversionrepair locality
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The pith

Lower bounds prove that existing constructions of locally repairable convertible codes achieve minimal bandwidth during global merges.

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

The paper derives the first non-trivial lower bounds on the total data transferred when converting between systematic optimal-distance locally repairable codes in the global merge regime. It generalizes an information-theoretic model of conversion to account for local repair properties and applies it to stable convertible codes that maximize the number of nodes left unchanged. These bounds hold without assuming the codes are linear and match the performance of prior constructions across a wide range of parameters. A sympathetic reader would care because distributed storage systems can then adjust their redundancy levels to match changing disk failure rates while using the least possible network bandwidth. If the bounds are tight, efficient conversion becomes feasible without hidden extra costs.

Core claim

We derive the first non-trivial lower bounds on the bandwidth cost of conversion between systematic optimal-distance Locally Repairable Codes (LRCs) in the global merge regime for stable convertible codes. These bounds hold without any linearity assumptions and match the performance of the constructions by Maturana and Rashmi across a broad range of parameters, establishing their bandwidth optimality.

What carries the argument

The generalized information-theoretic model of code conversion bandwidth, applied to stable convertible codes that preserve information locality when multiple initial LRC codewords merge into one final codeword.

If this is right

  • The constructions of Maturana and Rashmi achieve the minimal possible conversion bandwidth for a broad range of parameters.
  • The lower bounds apply to arbitrary codes, linear or nonlinear.
  • Redundancy adaptation in storage systems can occur with provably minimal data transfer while keeping local repair intact.
  • No conversion method can beat the identified bandwidth cost in this regime.

Where Pith is reading between the lines

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

  • The information-theoretic approach might be adapted to derive bounds for local-merge or other conversion regimes.
  • Storage architects could select these optimal convertible LRCs to minimize network load during dynamic redundancy changes.
  • Empirical measurements of actual conversion traffic in deployed systems could test whether the predicted savings materialize.

Load-bearing premise

The analysis is restricted to the global merge regime with stable convertible codes that maximize the number of unchanged nodes while preserving locality.

What would settle it

A conversion procedure between optimal-distance LRCs in the global merge regime that transfers strictly less data than the derived lower bound would falsify the optimality claim.

Figures

Figures reproduced from arXiv: 2604.15282 by K.V. Rashmi, Saransh Chopra, Shubhransh Singhvi.

Figure 1
Figure 1. Figure 1: A codeword of a (k = 9, g = 3, r = 3, ℓ = 2)-LRC. Empty boxes denote information symbols. L and G denote local parity symbols and global parity symbols, respectively. Moreover, there has recently been a growing interest in wide codes [8]–[10], i.e., codes with large dimension k, as they can achieve lower storage overhead for a prescribed level of reliability, further exacerbating repair costs of MDS-coded … view at source ↗
Figure 2
Figure 2. Figure 2: Conversion from a (k I = 9, gI = 3, r = 3, ℓ = 1)-LRC to a (k F = 18, gF = 4, r = 3, ℓ = 1)-LRC. Empty boxes denote information symbols. L and G denote local parity symbols and global parity symbols, respectively. We restrict our focus to stable convertible codes, which maximize the number of symbols that remain unchanged during the conversion procedure. For this regime, Maturana and Rashmi proposed constr… view at source ↗
Figure 3
Figure 3. Figure 3: A codeword of a (k = 9, g = 3, r = 3, ℓ = 2)-LRC. The notation for the random variables corresponding to the data stored in information and local parity nodes are globally indexed. k + µℓ + g nodes, where µ := k/r. These consist of k information nodes, each storing a message symbol directly, and µℓ + g parity nodes, storing deterministic functions of the message symbols. The k information nodes are divided… view at source ↗
Figure 4
Figure 4. Figure 4: Stable optimal-distance (k I = 9, gI = 3, r = 3, ℓ = 1; k F = 18, gF = 4, r = 3, ℓ = 1)-LRCC. We adopt a global indexing scheme for both the initial and final codes. Moreover, since the local groups remain unchanged, their labels are omitted in the final code for brevity. parity nodes (see [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

Recent studies have shown that distributed storage systems can achieve significant space savings by adapting redundancy levels to varying disk failure rates. This adaptation is performed via code conversion, wherein data encoded under an initial code are transformed to data encoded under a final code. While this process is typically resource-intensive, convertible codes are designed to enable these transformations efficiently while preserving desirable decodability constraints such as repair degree, or the number of nodes accessed during node repair. In this work, we focus on the bandwidth cost of conversion, or the total amount of data transferred during the conversion process. We study fundamental limits on the bandwidth cost of conversion between systematic optimal-distance Locally Repairable Codes (LRCs). We restrict our focus to the global merge regime, in which multiple initial codewords are combined to form a single final codeword while preserving information locality. We focus on stable convertible codes, wherein the number of unchanged nodes is maximized during conversion. We generalize an information-theoretic approach for modeling code conversion to the LRC setting, and derive the first non-trivial lower bounds on the bandwidth cost of conversion in this regime. Notably, our bounds do not rely on any linearity assumptions. Consequently, we show that the constructions of Maturana and Rashmi are bandwidth-optimal across a broad range of parameters in the global merge regime.

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

2 major / 2 minor

Summary. The paper claims to derive the first non-trivial information-theoretic lower bounds on conversion bandwidth for stable locally repairable convertible codes (LRCs) between systematic optimal-distance codes in the global merge regime. It generalizes prior conversion analysis to the LRC setting without linearity assumptions, focusing on stable codes that maximize unchanged nodes while preserving information locality, and concludes that the Maturana-Rashmi constructions achieve these bounds (hence are optimal) across a broad parameter range.

Significance. If the bounds and optimality results hold, this provides the first fundamental limits on bandwidth cost for LRC code conversion, which is relevant for adaptive redundancy in distributed storage. The non-linearity of the bounds and the explicit scoping to stable global-merge convertible codes strengthen applicability, while confirming optimality of known constructions offers immediate design guidance.

major comments (2)
  1. [Section 3] The generalization of the information-theoretic model (abstract and Section 3) incorporates LRC locality constraints into the conversion bandwidth lower bound; however, it is unclear whether the entropy accounting for repair-degree access during the merge fully captures all possible stable conversion strategies or introduces looseness that would affect the claimed tightness against the Maturana-Rashmi constructions.
  2. [Theorem 1 (or equivalent)] Theorem establishing the lower bound (likely Section 4) should explicitly delineate the parameter regime (e.g., in terms of locality parameters r, n, k and merge factors) where the bound is non-trivial and where the constructions meet it, to substantiate the abstract's claim of optimality 'across a broad range of parameters.'
minor comments (2)
  1. Notation for initial versus final code parameters and the definition of 'unchanged nodes' in stable conversion could be summarized in a table for clarity.
  2. [Abstract] The abstract states the bounds are 'non-trivial' but does not preview the key technical step (e.g., the cut or entropy inequality used); a one-sentence hint would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We have revised the manuscript to address the concerns about model completeness and to explicitly delineate the parameter regimes, thereby strengthening the presentation of our results.

read point-by-point responses

  1. Referee: [Section 3] The generalization of the information-theoretic model (abstract and Section 3) incorporates LRC locality constraints into the conversion bandwidth lower bound; however, it is unclear whether the entropy accounting for repair-degree access during the merge fully captures all possible stable conversion strategies or introduces looseness that would affect the claimed tightness against the Maturana-Rashmi constructions.

    Authors: The information-theoretic model in Section 3 is derived specifically for stable convertible codes in the global merge regime, where the entropy lower bound accounts for the minimum data movement required to satisfy both the final code's locality constraints and the stability condition (maximizing unchanged nodes). The accounting uses subadditivity and the locality parameter r to bound the information that must be transferred from changed nodes; any stable strategy must respect these entropy inequalities, so the bound captures the fundamental cost without assuming linearity. Tightness is established by explicit matching to the Maturana-Rashmi constructions, which achieve equality for the considered parameters. To remove any ambiguity, we have added a clarifying paragraph in the revised Section 3 that walks through why alternative stable strategies cannot undercut the bound. revision: yes

  2. Referee: [Theorem 1 (or equivalent)] Theorem establishing the lower bound (likely Section 4) should explicitly delineate the parameter regime (e.g., in terms of locality parameters r, n, k and merge factors) where the bound is non-trivial and where the constructions meet it, to substantiate the abstract's claim of optimality 'across a broad range of parameters.'

    Authors: We agree that an explicit delineation improves readability. In the revised manuscript we have inserted a new remark immediately following Theorem 1 that states the precise regime: the bound is non-trivial whenever the merge factor m satisfies m ≥ 2 and the locality parameters obey r < k with the optimal-distance condition d = n - k - ⌈k/r⌉ + 2 preserved under merging; equality with the Maturana-Rashmi constructions holds for all such parameters in the stable global-merge setting (i.e., the full range claimed in the abstract). A short table summarizing the regimes for representative (r, k, m) values has also been added for clarity. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior constructions; core lower bounds derived independently

full rationale

The paper generalizes an existing information-theoretic conversion model to the LRC setting and derives new lower bounds on bandwidth cost that do not rely on linearity or on the specific constructions being evaluated. These bounds are then used to establish optimality of the Maturana-Rashmi constructions for a range of parameters. While the cited constructions come from prior work by one of the present authors, the load-bearing step (the new lower-bound derivation) is presented as an independent generalization and does not reduce to a self-citation or to a fitted parameter renamed as a prediction. No self-definitional, ansatz-smuggling, or renaming patterns are visible in the stated approach.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; no explicit free parameters, invented entities, or detailed axioms listed. The central modeling step is a generalization of an information-theoretic approach.

axioms (1)
  • domain assumption Generalized information-theoretic model for code conversion applies to LRCs while preserving locality
    Paper states it generalizes an information-theoretic approach for modeling code conversion to the LRC setting.

pith-pipeline@v0.9.0 · 5538 in / 1197 out tokens · 47521 ms · 2026-05-10T09:43:54.200419+00:00 · methodology

discussion (0)

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