{"id":"77e50950-931d-4aa1-9fc7-05966e46a5be","arxiv_id":"2607.03857","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Function-correcting codes are extended to the sum-rank metric, with a Plotkin-like bound on irregular-distance codes and optimal-redundancy constructions for locally binary and weight functions.","lead":"The paper defines function-correcting codes under the sum-rank metric and proves a Plotkin-style lower bound on their redundancy. It also gives explicit optimal constructions for locally binary functions and sum-rank weight functions, useful for multi-shot network coding.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the matching constructions for locally binary and weight functions as the paper’s strongest, fully rigorous contribution. The only external ingredient (constant MRD codes of size q^m) is classical and is used only for the general Plotkin bound and for one regime of the weight-function construction; it does not affect the optimality statements that form the strongest claim. Consequently the reader’s ACCEPT verdict stands without adjustment. The suggested concrete test simply re-checks the elementary distance calculation that underpins Lemma 1, confirming that no algebraic subtlety has been overlooked.","tokens_in":17110,"tokens_out":514,"duration_ms":71924,"concrete_test":"Independently verify Lemma 1 for the concrete parameters of Example 6 (q=2,m=2,t=1,k=2): enumerate all pairs u,u' with f(u)\neq f(u') and d_srk(u,u')≤2, confirm that the single-block parity (I_2 or 0_2) forces d_srk(Enc(u),Enc(u'))≥3, and check that no shorter parity works. The same check for the weight-function case of Example 7 (r=1) settles the matching claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim (optimal redundancy exactly ceil(2t/m) for 2t-sum-rank locally binary functions via Construction 1/Lemma 1, and the same for sum-rank weight functions under the stated field-size conditions via Construction 2/Lemma 2) is self-contained. The lower bound follows immediately from the elementary N_srk(2,2t)=ceil(2t/m) (Corollary 1) that uses only the existence of a single full-rank matrix; the matching upper bound is realized by an explicit repetition of the identity/zero matrices (or of a constant-MRD set of size at most 2t+1). The classical existence of [m,1,m] MRD codes (Definition 4) is invoked only for the general Plotkin-like bound (Theorems 3–4) and for the weight-function construction when q^m is small; it is not required for the locally-binary optimality statement that the reader highlights. No hidden restriction on q or m undermines the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","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.","tokens_in":17355,"tokens_out":1081,"duration_ms":77253,"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":[{"comment":"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.","section":null},{"comment":"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.","section":null}],"minor_comments":[{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a clean, well-scoped extension of the FCC literature to the sum-rank metric. The two optimality results that the reader highlights are solid; the only load-bearing technical points that need a little extra care are the looseness of the weight-sum estimate in Theorem 4 and the edge-case verification in Construction 2. I see no novelty or citation issues. Fit for a solid IT journal is good after the minor revisions above."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is the first treatment of function-correcting codes under the sum-rank metric. The new pieces are a Plotkin-like lower bound for irregular sum-rank distance codes (Theorem 3), its specialization to linear functions (Theorem 4), and two explicit constructions that achieve the elementary lower bound ceil(2t/m) for 2t-sum-rank locally binary functions and for sum-rank weight functions (under the usual field-size conditions).\n\nWhat works: the proofs are standard double-counting plus the known existence of constant MRD codes of size q^m. The bound recovers the Hamming and rank-metric Plotkin statements as special cases, which is a useful sanity check. Construction 1 is especially clean: just repeat the identity or zero matrix enough times; the matching lower bound needs only a single full-rank matrix, so the optimality claim for locally binary functions does not even depend on the full MRD machinery. The weight-function construction is a bit more bookkeeping when q^m = 2t, but the cases are written out and the distance calculations close.\n\nSoft spots are minor and proportional. A few arguments are waved through as “along the same lines as the Hamming case,” and the linear-function Plotkin bound uses the usual weight-sum inequality via the field isomorphism, which is slightly loose but not load-bearing. No circularity, no free parameters, no invented objects beyond the natural DRM for the new metric. Citation pattern is appropriate: the FCC literature plus the standard sum-rank references.\n\nThis is for people already working on sum-rank codes or multi-shot network coding who want function protection with less redundancy. It is not a breakthrough outside that circle, but it is honest progress and formally grounded enough that a serious editor should send it to referees. I would cite the bound and the two constructions if I needed them; I would bring it to reading group only if we are already deep in sum-rank or FCC papers that week. Worth peer review.","headline":"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.","tokens_in":17956,"tokens_out":505,"would_cite":true,"duration_ms":62546,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B65","94B60"],"pacs":[],"model":"grok-4.5","headline":"Function-correcting codes under the sum-rank metric achieve optimal redundancy ceil(2t/m) for locally binary and weight functions.","keywords":["function-correcting codes","sum-rank metric","Plotkin-like bound","optimal redundancy","locally binary functions","sum-rank weight","irregular-distance codes","MSRD codes"],"falsifier":"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.","tokens_in":18023,"feed_emoji":"📡","tokens_out":934,"duration_ms":90850,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sum-rank function codes hit optimal length ceil(2t/m)","feed_subtitle":"Locally binary and weight functions need only that many matrix blocks to survive t errors","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Sum-rank FCCs achieve optimal redundancy exactly ceil(2t/m)","Locally binary functions need only ceil(2t/m) matrix blocks","Irregular-distance codes determine sum-rank FCC redundancy","Constant-rank repetition yields optimal sum-rank FCC length","Plotkin-like bound settles optimal FCC redundancy in sum-rank"],"cache_read_input_tokens":14080,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sum-rank FCCs achieve optimal redundancy exactly ceil(2t/m)","Locally binary functions need only ceil(2t/m) matrix blocks","Irregular-distance codes determine sum-rank FCC redundancy","Constant-rank repetition yields optimal sum-rank FCC length","Plotkin-like bound settles optimal FCC redundancy in sum-rank"]},"model":"grok-4.5","effort":"low","cost_usd":0.00401,"raw_usage":{"total_tokens":1201,"prompt_tokens":703,"num_sources_used":0,"completion_tokens":93,"cost_in_usd_ticks":40100000,"prompt_tokens_details":{"text_tokens":703,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":405,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":703,"tokens_out":93,"duration_ms":3592,"temperature":1.0,"reasoning_tokens":405,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T07:03:41.887327+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":2}