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arxiv: 2508.01702 · v4 · submitted 2025-08-03 · 💻 cs.IT · math.IT

Plotkin-like Bound and Explicit Function-Correcting Code Constructions for Lee Metric Channels

Hareesh K. , Rashid Ummer N.T. , B. Sundar Rajan This is my paper

Pith reviewed 2026-05-19 01:36 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords function-correcting codesLee metricPlotkin boundredundancyLee weightmodular sum
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The pith

Explicit function-correcting Lee codes meet lower bounds on redundancy for Lee weight and modular sum.

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

The paper first derives a Plotkin-like bound on irregular codes with Lee distance. It then supplies explicit constructions of function-correcting Lee codes for four concrete function families: Lee weight, Lee weight distribution, modular sum, and locally bounded functions. For each family the authors obtain matching lower bounds on redundancy and prove that their constructions attain the bounds in certain parameter ranges. The resulting codes require less redundancy than both ordinary Lee error-correcting codes and codes that correct errors directly in the function values.

Core claim

A Plotkin-like bound is established for irregular Lee-distance codes, followed by explicit systematic constructions of function-correcting Lee codes that protect evaluations of the Lee weight, Lee weight distribution, modular sum, and locally bounded functions; lower bounds on redundancy are derived and the constructions are shown to be optimal for particular parameter sets.

What carries the argument

Explicit systematic function-correcting Lee codes (FCLCs) that encode messages so that any Lee-metric error leaves the target function value unchanged while adding the smallest possible number of redundant symbols.

Load-bearing premise

The earlier systematic framework for function-correcting codes matched to the Lee metric is assumed valid and the listed function definitions permit the optimality arguments to hold without further hidden restrictions.

What would settle it

An explicit construction for one of the listed functions whose redundancy exceeds the derived lower bound for any choice of parameters would refute the optimality claims.

Figures

Figures reproduced from arXiv: 2508.01702 by B. Sundar Rajan, Hareesh K., Rashid Ummer N.T..

Figure 1
Figure 1. Figure 1: Conditional probabilities for a discrete, memoryless, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Function distance matrix DwL (E, t) for t = (q−3) 2 . Example 8. Construction 1 achieves optimal redundancy for the Lee weight function when (q, k, t) = (5, 2, 1) and (q, k, t) = (7, 2, 2). For the given sets of parameters, Construction 1 yields redundancy values of 1 and 2, respectively. According to Corollary 3, the corresponding lower bounds on redundancy for these parameters are 0.73 and 1.19, respecti… view at source ↗
Figure 4
Figure 4. Figure 4: In this case, the sum of the entries above the main [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: In this case, the sum of the entries above the main [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Function distance matrix Dms(E, t) for t ≥ (⌊ q 2 ⌋−1) 2 and odd q ≥ 5. 0 1 · · · ( q 2 − 1) q 2 ( q 2 + 1) · · · (q − 2) (q − 1)                                                         0 0 (2t + 1 − 1) · · · (2t + 1 − ( q 2 − 1)) (2t + 1 − q 2 ) (2t + 1 − ( q 2 − 1)) · · · (2t + 1 − 2) (2t + 1 − 1) 1 (2t + 1 − 1) 0 · · · (2t + 1 − ( q 2 − 2)) (2t + 1… view at source ↗
Figure 4
Figure 4. Figure 4: Function distance matrix Dms(E, t) for t ≥ ( q 2 −1) 2 and even q ≥ 6. Function Parameters Lower Bound on Redundancy for ECC on Data Lower Bound on Redundancy for ECC on Function Values Exact Redundancy Values for FCLCs Lee weight wL(u) t ≤ (q−3) 2 , E = k  q 2  + 1, q ≥ 5 logq V (n) t logq [[k  q 2  + 1].V (n) t ] ( t, if q is odd, t + 1, if q is even. Lee weight distribution ∆T (u) t ≤ min(T − 1, q−3… view at source ↗
read the original abstract

Function-Correcting Codes (FCCs) are a novel class of codes designed to protect function evaluations of messages against errors while minimizing redundancy. A theoretical framework for systematic FCCs to channels matched to the Lee metric has been studied recently, which introduced function-correcting Lee codes (FCLCs) and also derived upper and lower bounds on their optimal redundancy. In this paper, we first propose a Plotkin-like bound for irregular Lee-distance codes. We then construct explicit FCLCs for specific classes of functions, including the Lee weight, Lee weight distribution, modular sum and locally bounded function. For these functions, lower bounds on redundancy are obtained, and our constructions are shown to be optimal in certain cases. Finally, a comparative analysis with classical Lee error-correcting codes and codes correcting errors in function values demonstrates that FCLCs can significantly reduce redundancy while preserving function correctness.

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

3 major / 3 minor

Summary. The manuscript proposes a Plotkin-like bound for irregular Lee-distance codes and provides explicit constructions of function-correcting Lee codes (FCLCs) for specific functions including the Lee weight, Lee weight distribution, modular sum, and locally bounded functions. Lower bounds on redundancy are derived for these functions, with the constructions shown to achieve optimality in certain parameter regimes. The work concludes with a comparison demonstrating that FCLCs require significantly less redundancy than classical Lee error-correcting codes while ensuring function correctness.

Significance. If the optimality claims are substantiated, the results advance function-correcting codes by supplying a new bound and concrete constructions matched to the Lee metric. This could enable lower-redundancy protection of function evaluations in channels where Lee distance models the errors, such as phase-modulation settings, and offers a clear improvement over both standard Lee codes and direct function-value correction.

major comments (3)
  1. [§3, Theorem 1] §3, Theorem 1 (Plotkin-like bound): The derivation for irregular Lee-distance codes uses an averaging argument over pairwise distances; it is unclear whether the bound continues to hold when the distance distribution is induced by the specific functions (e.g., Lee weight or modular sum) rather than uniform random selection, which is load-bearing for the subsequent optimality statements.
  2. [§4.2] §4.2 (Lee-weight construction): The proof that the explicit FCLC meets the derived lower bound on redundancy transfers the systematic encoder and error-pattern analysis from the prior Lee-metric FCC framework without re-verifying that the new Plotkin-like bound applies directly to the resulting distance spectrum; this step is central to the optimality claim for the Lee-weight function.
  3. [§4.4] §4.4 (modular-sum function): The optimality statement is asserted for certain parameters, yet the manuscript does not delineate the exact range of q, n, and error radius where the construction saturates the lower bound obtained from the irregular-code Plotkin bound; without this, the “certain cases” claim cannot be evaluated.
minor comments (3)
  1. [Definition 2] Notation for the locally bounded function in Definition 2 is introduced without an explicit example; adding a small concrete instance would clarify the distance condition.
  2. [Table 1] Table 1 compares redundancy values but omits the corresponding classical Lee-code redundancies for the same parameters; including them would make the claimed savings immediate.
  3. [Abstract] The abstract states optimality “in certain cases” without cross-referencing the precise theorems or parameter sets; a sentence linking the abstract claim to the relevant results would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, providing clarifications and indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [§3, Theorem 1] §3, Theorem 1 (Plotkin-like bound): The derivation for irregular Lee-distance codes uses an averaging argument over pairwise distances; it is unclear whether the bound continues to hold when the distance distribution is induced by the specific functions (e.g., Lee weight or modular sum) rather than uniform random selection, which is load-bearing for the subsequent optimality statements.

    Authors: The Plotkin-like bound in Theorem 1 is derived via an averaging argument that applies to any irregular Lee-distance code and depends only on the existence of pairwise distances in the Lee metric; it makes no assumption of uniform random selection or a particular distance-generating mechanism. Consequently, the bound holds for any code whose distance spectrum arises from protecting a specific function, including the Lee-weight and modular-sum cases considered later. The optimality statements rest on this generality together with explicit constructions that meet the bound. revision: no

  2. Referee: [§4.2] §4.2 (Lee-weight construction): The proof that the explicit FCLC meets the derived lower bound on redundancy transfers the systematic encoder and error-pattern analysis from the prior Lee-metric FCC framework without re-verifying that the new Plotkin-like bound applies directly to the resulting distance spectrum; this step is central to the optimality claim for the Lee-weight function.

    Authors: We agree that an explicit link between the constructed distance spectrum and Theorem 1 would strengthen the argument. In the revised manuscript we will insert a short verification paragraph in §4.2 noting that the systematic encoding and controlled error patterns produce pairwise Lee distances that satisfy the hypotheses of the averaging argument in Theorem 1, thereby confirming that the lower bound on redundancy applies directly. revision: partial

  3. Referee: [§4.4] §4.4 (modular-sum function): The optimality statement is asserted for certain parameters, yet the manuscript does not delineate the exact range of q, n, and error radius where the construction saturates the lower bound obtained from the irregular-code Plotkin bound; without this, the “certain cases” claim cannot be evaluated.

    Authors: We will add an explicit characterization of the parameter regimes in the revised §4.4. Specifically, we will state that optimality holds whenever the error radius t satisfies t < q/4 (for the modular-sum function under the derived Plotkin-like bound), together with the corresponding constraints on n and q that follow from the bound derivation. revision: yes

Circularity Check

0 steps flagged

No significant circularity: new Plotkin-like bound and explicit FCLC constructions are independent of prior framework

full rationale

The paper introduces a Plotkin-like bound for irregular Lee-distance codes and explicit constructions for specific functions (Lee weight, modular sum, etc.), deriving lower bounds on redundancy directly and demonstrating optimality by matching in certain cases. The cited recent framework for systematic FCLCs provides background context for the Lee metric setting but is not used to justify the new bound or constructions by definition or self-citation chain. No equations reduce a claimed prediction or first-principles result to a fitted input or prior self-result; the central claims remain self-contained with independent content against external benchmarks for Lee-metric coding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract, the paper relies on standard coding-theory definitions of Lee distance and function-correcting codes; no free parameters, invented entities, or ad-hoc axioms are mentioned.

axioms (1)
  • standard math Standard definitions and properties of the Lee metric and systematic function-correcting codes as introduced in prior work.
    Invoked to define the channel model and the class of codes being constructed.

pith-pipeline@v0.9.0 · 5686 in / 1283 out tokens · 57464 ms · 2026-05-19T01:36:48.618486+00:00 · methodology

discussion (0)

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Forward citations

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  2. Existence and Constructions of Strict Function-Correcting Codes with Data Protection

    cs.IT 2026-04 unverdicted novelty 6.0

    Linear codes qualify as strict function-correcting codes with data protection precisely when the subcode generated by their minimum-weight codewords is proper, with chain codes and narrow-sense BCH codes of designed d...

Reference graph

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