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REVIEW 2 major objections 5 minor 21 references

Function-correcting codes under the sum-rank metric achieve optimal redundancy ceil(2t/m) for locally binary and weight functions.

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2026-07-13 07:03 UTC pith:HL7VAWW5

load-bearing objection Clean first extension of FCCs to sum-rank metric with a usable Plotkin-style bound and two matching optimal constructions; solid, limited-scope theory paper. the 2 major comments →

arxiv 2607.03857 v2 pith:HL7VAWW5 submitted 2026-07-04 cs.IT math.IT

Function-Correcting Codes for Sum-Rank Metric

Santhi Kumari Kammila , B. Sundar Rajan This is my paper
classification cs.IT math.IT MSC 94B0594B6594B60
keywords function-correcting codessum-rank metricPlotkin-like boundoptimal redundancylocally binary functionssum-rank weightirregular-distance codesMSRD codes
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Function-correcting codes protect a function of a message more carefully than the full message itself, using far less redundancy than ordinary error-correcting codes. This paper carries that idea into the sum-rank metric, the natural setting for multi-shot network coding and distributed storage. It gives a general Plotkin-style lower bound on the extra length needed for irregular-distance codes, recovers the classical Hamming and rank-metric Plotkin bounds as special cases, and then specializes the bound to linear functions. For two concrete families—sum-rank locally binary functions and the sum-rank weight function—the paper supplies explicit parity constructions whose redundancy is exactly the absolute lower bound ceil(2t/m). A reader who cares about reliable recovery of simple attributes (parity, weight, local decisions) over matrix-valued channels therefore obtains both a theoretical floor and matching codes that meet it.

Core claim

Under the sum-rank metric the optimal redundancy of a function-correcting code is completely determined by the smallest length of an irregular-distance code whose pairwise distances meet the function-dependent distance-requirement matrix. For every 2t-sum-rank locally binary function, and for the sum-rank weight function whenever q^m is large enough relative to t, that optimal redundancy equals exactly ceil(2t/m) and is achieved by a simple constant-rank repetition construction.

What carries the argument

The distance-requirement matrix Df_srk together with the associated irregular-sum-rank-distance codes N_srk(D). Pairwise sum-rank distances among the parity vectors are forced to satisfy the entries of Df_srk; the Plotkin-like counting argument then converts those distance lower bounds into a lower bound on the number of matrix blocks r.

Load-bearing premise

The Plotkin counting argument assumes that one can pack q^m matrices of size m by m so that every pair differs by full rank m; without that constant-rank set the numerical bound on redundancy need not hold.

What would settle it

Fix q, m, t with q^m >= 2t and construct the constant-rank set of size q^m. Build the claimed parity vectors of length ceil(2t/m) for the sum-rank weight function and check whether every pair of messages with distinct weights produces encoded vectors whose sum-rank distance is at least 2t+1; any counter-example pair falsifies optimality.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any 2t-sum-rank locally binary function can be protected against t sum-rank errors by appending only ceil(2t/m) matrix blocks.
  • The sum-rank weight function itself admits an optimal FCSRC of the same length whenever the ambient field is large enough.
  • The new Plotkin-like bound recovers both the classical Hamming Plotkin bound (m=1) and the known sum-rank Plotkin bound for ordinary codes, giving a single counting argument that unifies the three metrics.
  • Linear functions receive a sharpened numerical lower bound that subtracts the kernel weight sum, immediately usable for any full-rank matrix representation of the function.

Where Pith is reading between the lines

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

  • The same constant-rank building block can be reused to produce optimal FCSRCs for other locally constant functions whose image balls have size at most q^m.
  • Because the sum-rank metric interpolates Hamming and rank metrics, every new function class solved for sum-rank automatically yields matching results for the two classical metrics by setting m=1 or the number of blocks to one.
  • Multi-shot network-coding applications that already employ sum-rank codes can replace full-message protection by these lighter function-correcting codes whenever only a low-cardinality attribute is required at the decoder.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces function-correcting codes for the sum-rank metric (FCSRCs). It defines the associated distance-requirement matrix and irregular-sum-rank-distance codes, proves that the optimal redundancy equals N_srk of that matrix (Theorem 1), and derives a Plotkin-like lower bound on N_srk(D) by pairwise rank counting that uses the existence of a constant-rank-distance set of size q^m (Theorem 3). The bound is specialized to linear functions (Theorem 4). Explicit constructions are given for 2t-sum-rank locally binary functions (Construction 1 / Lemma 1) and for the sum-rank weight function under the condition q^m >= 2t (Construction 2 / Lemma 2); both achieve the elementary lower bound ceil(2t/m) and are therefore optimal for those classes.

Significance. Sum-rank metric codes are the natural model for multi-shot network coding and certain distributed-storage settings. Extending the FCC framework of Lenz et al. to this metric, recovering the known Hamming and rank-metric Plotkin bounds as special cases, and supplying matching constructions for two natural function classes is a solid incremental contribution. The optimality statements for locally binary and weight functions rest only on elementary distance arguments and the existence of a single full-rank matrix (or a small constant-MRD set), so they are robust. The general Plotkin-like bound and its linear specialization supply a reusable tool for future work on other functions.

major comments (2)
  1. Theorem 4 (linear-function Plotkin bound): the derivation replaces sum_u w_srk(u) by the larger Hamming-weight sum via the column-wise isomorphism Phi, producing a valid but possibly loose lower bound on r. The paper should either compute (or bound more tightly) the exact sum of sum-rank weights over (F^{m x m}_q)^k, or explicitly state that the resulting expression is only a lower bound that may not be tight. Examples 4 and 5 already show a gap between the two sums; quantifying its effect on the final r would strengthen the claim.
  2. Construction 2 / Lemma 2 (sum-rank weight functions): when q^m = 2t the construction introduces an auxiliary rank-1 matrix E_m and claims d_srk(p_{2t}, p_{2t+1}) = 2t. The case analysis that follows is lengthy and relies on several ad-hoc distance calculations (especially Case 2b). A short explicit verification that the chosen E_m always yields the required distances for every pair, or a reference to a systematic way of completing a constant-MRD set of size 2t to size 2t+1, would make the optimality claim fully self-contained.
minor comments (5)
  1. Several proofs (Theorems 1 and 2) are declared "along the same lines" as the Hamming-metric arguments of [1] without restating the short steps. Adding a one-sentence sketch for each would improve readability for readers unfamiliar with the original FCC paper.
  2. Notation: the manuscript mixes F_q^{m x m} and (F^{m x m}_q)^k; a single consistent convention would help. Likewise, the identity and zero matrices are written both Im / 0m and I_m / 0_m.
  3. Typographical issues: "distnace", "fucntions", "Futher", "overb-symbol", "sum-rank weight function is defined as f(u)=w_srk(u) where … and m,k in N" (missing punctuation), and a few missing spaces after commas in the abstract and introduction.
  4. Definition 4 cites [5] for the existence of [m,1,m] constant MRD codes; a one-line pointer to the Gabidulin construction (or to the symmetric-matrix construction when char = 2) would make the paper more self-contained.
  5. In the abstract the phrase "for locally binary functions with optimal redundancy" is slightly incomplete; the body treats both locally binary and sum-rank weight functions. Aligning the abstract with the contributions list would avoid confusion.

Circularity Check

0 steps flagged

No significant circularity: bounds and optimal constructions are derived from first principles and explicit encodings, not from self-referential fits or load-bearing self-citations.

full rationale

The paper introduces FCSRCs by direct analogy with the Hamming-metric FCC framework of [1], then derives a Plotkin-like lower bound (Theorem 3) by elementary pairwise sum-rank counting that maximises the contribution of each coordinate via a constant-rank-distance set of size q^m (Definition 4). The linear-function specialisation (Theorem 4) follows by the same counting restricted to cosets of ker(f). Both constructions that achieve the claimed optimal redundancy ceil(2t/m) are fully explicit: Construction 1 simply repeats the identity or zero matrix according to the local maximum of a 2t-sum-rank-locally-binary function, and Construction 2 assigns a short list of constant-MRD parity vectors (or a mild modification when q^m = 2t) according to the integer value of the sum-rank weight. The matching lower bound is the elementary N_srk(2,2t)=ceil(2t/m) of Corollary 1, which uses only the existence of a single full-rank matrix. Classical MRD existence results are external, parameter-free, and not required for the locally-binary optimality statement. No quantity is fitted to data and then re-presented as a prediction; no uniqueness theorem is imported from the authors’ own prior work; and the derivation chain does not reduce any claimed result to its own definitional inputs. Score 0 is therefore the correct assessment.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The paper rests on standard finite-field linear algebra, the definition of the sum-rank metric, and the classical existence of constant-rank MRD codes of size q^m. No numerical parameters are fitted; the only “invented” objects are the natural extensions of distance-requirement matrices and D-codes to the sum-rank setting, which are definitional rather than ontological.

axioms (3)
  • domain assumption Existence of an [m,1,m] constant maximum-rank-distance code of size q^m over F_{q^m} (Definition 4, used in the Plotkin counting argument and in Construction 2).
    Invoked without proof; existence follows from Gabidulin’s classical construction, which is external to the paper.
  • standard math The sum-rank weight is a metric (Definition 2).
    Standard fact from the sum-rank literature; used throughout distance arguments.
  • standard math For any function with at least two values, N_srk(2,2t)=ceil(2t/m) (Corollary 1).
    Immediate from the definition of sum-rank distance and the maximal rank of an m imes m matrix.
invented entities (1)
  • Function-correcting sum-rank code (FCSRC) and the associated distance-requirement matrix D^f_srk independent evidence
    purpose: To formalize the minimal redundancy needed to protect a function under sum-rank errors.
    Direct translation of the Hamming-metric FCC framework; no new physical or algebraic object is postulated beyond the metric change.

pith-pipeline@v1.1.0-grok45 · 21205 in / 2312 out tokens · 94087 ms · 2026-07-13T07:03:41.887327+00:00 · methodology

0 comments
read the original abstract

Function-Correcting Codes (FCCs) are a class of codes designed to protect the evaluation of a specific function of a message against channel errors at a higher level than the level of protection for the message, while requiring significantly less redundancy than conventional error-correcting codes. In this paper, we study function-correcting codes under the sum-rank metric, which is a natural generalization of both the Hamming metric and the rank-metric and also we derive general upper and lower bounds on the optimal redundancy of FCCs in the sum-rank metric. In particular, we establish a Plotkin-like bound for irregular-distance codes in sum-rank metric. Furthermore, we present explicit construction of function-correcting sum-rank metric codes (FCSRCs) for locally binary functions with optimal redundancy.

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Reference graph

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