{"id":"eab54ed9-db21-415e-9be6-6a7353029547","arxiv_id":"2501.18989","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Appending one symbol to known optimal LRCs preserves optimality, yielding length q+2 over F_q and new optimal (r,3)-LRC families.","lead":"The paper extends known optimal locally repairable codes by adding a single coordinate, while keeping the locality and the optimal minimum distance. This yields optimal LRCs of length q+2 over F_q and new optimal (r,3)-LRCs from rational and elliptic function fields.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 13's M'_u invertibility condition is stronger than Lemma 4 and is never instantiated; without it the elliptic (r,3)-LRC construction may be vacuous.","rationale":"I reviewed Theorems 5, 6, 7, and 11 carefully; the rational-function-field extension and the one-coordinate elliptic extension appear coherent, and the corresponding optimality proofs are consistent with the Singleton-type bounds. The reader's weakest-assumption analysis correctly identifies Theorem 13 as the largest unresolved gap: the required (r−1)×(r−1) invertibility of M'_u is strictly stronger than the r×r property established in [22, Lemma 4], and the paper provides neither a proof nor an explicit construction satisfying it. This leaves a central claimed family of optimal (r,3)-LRCs from elliptic function fields without a verified instantiation. The Roth-Lempel section also lacks a formal optimality condition, but the elliptic (r,3) gap is more concrete and more load-bearing because the hypothesis may be unsatisfiable. Since the reader already returned CONDITIONAL and my analysis does not change that assessment, the verdict should remain unchanged.","tokens_in":63,"tokens_out":18157,"duration_ms":344294,"concrete_test":"For the smallest case covered by [22] (e.g., q=4, r=2, an elliptic curve over F_4 with automorphism group of order 3), explicitly compute the rational places, the function ω1 from Lemma 4, and the matrices M'_u for the splitting-completely blocks. Check whether every (r−1)×(r−1) submatrix of M'_u is invertible. If some block has a zero ω1 entry, test whether choosing a different Q0 or a different set of s blocks avoids all such entries; if no valid set exists, the hypothesis of Theorem 13 is unsatisfiable for that parameter set.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The construction of optimal (r,3)-LRCs over elliptic function fields in Section 5.2 rests on Lemma 12, which requires that for every block u and every choice of r−1 rows, the (r−1)×(r−1) submatrix of M'_u = [ω1(Pu,v),...,ω_{r−1}(Pu,v)]_{v=1}^{r+1} is invertible. This is strictly stronger than the property proved in [22, Lemma 4], which only asserts that every r×r submatrix of M_u = [1, ω1,...,ω_{r−1}] is invertible. For r=2, Lemma 4 gives three pairwise distinct values of ω1 on a block, but the new condition additionally requires all three values to be nonzero; a row with ω1=0 is allowed by Lemma 4 but would make a 1×1 submatrix singular and destroy the locality argument in Lemma 12. The paper neither proves that the Li–Ma–Xing automorphism-group construction yields such point sets nor gives any explicit example where the condition holds. Theorem 13 is therefore a conditional statement with no verified hypothesis; if no point set satisfies it, the claimed optimal (r,3)-elliptic LRC family is empty. This is the main load-bearing gap in the paper's elliptic section.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies optimal locally repairable codes (LRCs) constructed from function fields. It proposes a general extension mechanism: given an optimal LRC in which every repair block is an [r+1,r] MDS code, adding one coordinate to a block, or one coordinate to every block, preserves optimality or yields an optimal (r,3)-LRC. The main constructions cover rational function fields (extending the codes of Barg–Tamo–Vladut and Jin–Ma–Xing to length q+2), Roth–Lempel type extensions, and elliptic function fields (extending the codes of Li–Ma–Xing to length about q+2√q, plus a conditional (r,3)-elliptic construction). Theorem 5, Theorem 6, Theorem 7, and Theorem 11 contain the core extension arguments; Theorem 13 is conditional on an unproved matrix invertibility condition. Section 4 claims optimality for Roth–Lempel type extensions, but the formal statement in Corollary 9 proves only a weaker distance bound.","tokens_in":17685,"tokens_out":21633,"duration_ms":179857,"significance":"If the main theorems are correct, the extension idea is simple and useful: it increases the known maximal length of optimal LRCs by one coordinate in both the rational and elliptic cases, while preserving locality and optimality. The proofs of the rational-field results (Theorems 5–7) are largely checkable and constructive, and the paper gives explicit parameter examples. However, two load-bearing gaps remain. First, the elliptic (r,3)-LRC construction in Section 5.2 depends on a stronger-than-known invertibility condition that is never proved or instantiated. Second, the Roth–Lempel section claims optimality 'under certain conditions' without stating them, and Corollary 9 only establishes a distance bound one below the Singleton-type bound. These issues prevent the paper from being accepted in its present form, but they are local and may be repairable within the manuscript's scope.","major_comments":[{"comment":"The proof of locality for the appended coordinate a_{0,t} assumes that z^{t-1}(Q_1) is nonzero, since c_0 = a_{0,t} z^{t-1}(Q_1) must be divided by z^{t-1}(Q_1) to recover a_{0,t}. The function z has a zero somewhere in F, and if Q_1 is chosen to be that zero then the appended symbol is invisible in the first block, so locality fails. The theorem statement does not require z(Q_1) ≠ 0, nor does it prove that Q_1 can be chosen to avoid the zero of z while still having enough splitting-complete places. This condition must be added and its existence established for the length q+2√q−2r−2 claim to hold.","section":"Section 5.1, Theorem 11"},{"comment":"Lemma 12 requires that every (r−1)×(r−1) submatrix of M'_u is invertible. This is strictly stronger than the property proved in [22, Lemma 4], which only asserts that every r×r submatrix of M_u = [1, ω_1, …, ω_{r−1}] is invertible. For r=2, Lemma 4 allows three distinct values of ω_1, including a possible zero value, but the new condition additionally requires all three values to be nonzero. The paper neither proves that the automorphism-group construction yields such a point set nor gives an explicit example satisfying the condition. If no such set exists, the claimed optimal (r,3)-elliptic LRC family is empty, so Theorem 13 remains a conditional statement with an unverified hypothesis.","section":"Section 5.2, Lemma 12 and Theorem 13"},{"comment":"The claimed Roth–Lempel optimality is not established. Corollary 9 proves only d ≥ n−rt−t+1, which is one less than the Singleton-type bound d ≤ n−rt−t+2 for an [n, rt] LRC with locality r. The text after the proof says optimality holds 'if the elements satisfying some conditions', but those conditions are never stated. In addition, the code C_{RL,1} is defined with two appended coordinates (a_{r−2,t}, a_{r−1,t}), so its length should be s(r+1)+2, not s(r+1)+1 as written in Corollaries 9 and 10. The distance analysis must explicitly account for both appended symbols before any optimality claim can be made.","section":"Section 4, Corollary 9"},{"comment":"The stated length n=(s+1)(r+1) is inconsistent with the displayed definition of C^m_e. Counting the coordinates in its defining tuple gives s(r+1)+s (or more, depending on the intended appended symbols), not (s+1)(r+1). This inconsistency affects the validity of the claimed d-optimality, since the Singleton-type bound depends on n. The definition and the length must be corrected, and the optimality proof should be re-run with the correct parameters.","section":"Corollary 8"}],"minor_comments":[{"comment":"In the paragraph for r=3, the code is described as having 'locality 4', but the construction has locality r=3; this is a typo.","section":"Example 1"},{"comment":"The sentence 'by Theorem 7' should read 'by Theorem 6', since it refers to the rational function field length-q+2 construction.","section":"Example 2"},{"comment":"In the proof of Theorem 13, the text says 'the length and dimension of the code CE are indeed s(r+1)+1 and rt−r+1', but the code CE has length s(r+2); this should be corrected.","section":"Theorem 13 proof"},{"comment":"Condition 3 in Theorem 5 is stated as r+1 = u p^v; the paper should specify that u is an integer coprime to p, since otherwise the condition is ambiguous.","section":"Theorem 5"},{"comment":"The notation Q_∞, P_∞ in Section 5 is introduced without a clear connection to the places Q_0, P_{0,1}, …, P_{0,r+1} used in Section 2.5; aligning this notation would improve readability.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript overclaims in two places: the abstract promises optimal Roth–Lempel type LRCs, but no theorem in Section 4 proves optimality; and the abstract's 'exploring one more condition' for elliptic (r,3)-LRCs turns out to be a strong matrix condition that is never proved or exemplified. In addition to the technical fixes detailed in the major comments, the authors should revise the abstract and introduction to match the formally proven statements. The rational-function-field results appear sound and are the strongest part of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something useful and mostly correct. The core idea is a single-coordinate extension of optimal LRCs, modeled on extended Reed-Solomon codes. Applied to the known function-field constructions, it gives optimal LRCs over F_q of length q+2 when (r+1)|(q+1), one beyond the q+1 of Jin–Ma–Xing, and it also produces new optimal (r,3)-LRCs by extending every block. Theorems 5, 6, and 11 check out: the appended coordinate does not change the dimension, locality holds via interpolation on each block, and the minimum-distance calculation matches the Singleton-type bound. I did not find circular reasoning; the paper genuinely extends external constructions. The rational-function-field part is the main contribution and it appears sound.\n\nThe soft spots are real but localized. The elliptic (r,3)-LRC construction in Section 5.2 depends on Lemma 12's assumption that every (r−1)×(r−1) submatrix of M'_u is invertible. That is strictly stronger than the property proved in [22, Lemma 4], which only gives invertibility of every r×r submatrix of M_u. The paper neither proves this stronger condition nor gives a single point set that satisfies it. For r=2, the condition requires all three omega_1 values in a block to be nonzero, which Lemma 4 does not guarantee; a zero value would break the (r,3)-locality argument. So Theorem 13 is conditional on an unverified hypothesis, and the claimed optimal elliptic (r,3)-LRC family may be empty. This is a load-bearing gap in that section. The Roth-Lempel extension in Section 4 is also underdeveloped: the text promises optimality under \"some conditions\" but Corollary 9 only proves d ≥ n−rt−t+1, one below the Singleton bound, and the conditions are never stated precisely.\n\nThere are typos and notation slips throughout, but they do not obscure the main arguments. The reference list is appropriate; the paper cites the relevant prior constructions and does not overstate its novelty. The rational parts are new and correct, the elliptic (r,3) part is genuinely incomplete.\n\nWho this is for: anyone working on optimal LRC constructions from function fields, especially on length extension tricks. The rational-function-field results deserve to be in the literature. The paper deserves a serious referee: a careful referee can verify Theorems 5–7 and 11, and should push the authors to either prove or instantiate the M'_u condition for Theorem 13, or remove the claim. I would accept it for peer review with the expectation that the elliptic (r,3)-LRC section needs major revision or excision.","headline":"A solid incremental paper: the rational-function-field extension to length q+2 and the (r,3)-LRCs are real, but the elliptic (r,3)-LRC claim rests on an unproved and possibly false invertibility condition.","tokens_in":18254,"tokens_out":1270,"would_cite":false,"duration_ms":13461,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B27","11T71","14H52"],"pacs":[],"model":"deepseek-v4-flash","headline":"Extending one coordinate of an optimal LRC preserves optimality, pushing rational-field lengths to q+2 and improving elliptic-field LRCs.","keywords":["locally repairable codes","optimal LRC","function fields","elliptic curves","rational function fields","(r,3)-LRC","Singleton-type bound","extended codes"],"falsifier":"Take $q=64$ and an elliptic curve over $\\mathbb{F}_{64}$ with automorphism group of order $24$, choose $r=7$ and a block of points as in Section 2.5, and compute the $6\\times 6$ minors of the matrix $M'_u$; if any of these $(r-1)\\times(r-1)$ minors vanishes for a block that satisfies the $r\\times r$ condition of [22, Lemma 4], then the hypothesis of Theorem 13 fails and the constructed $(r,3)$-LRC is not guaranteed by the paper's proof.","tokens_in":1789,"feed_emoji":"","tokens_out":2466,"duration_ms":75885,"temperature":0.7,"pith_summary":"This paper claims that optimal locally repairable codes (LRCs) can be extended by a single coordinate attached to one repair block without losing either locality or optimality. For codes built from rational function fields, the extension pushes the maximal achievable length from q+1 to q+2 over $\\mathbb{F}_q$. For optimal LRCs from elliptic function fields, the same one-coordinate extension preserves optimality and yields lengths up to $q+2\\sqrt{q}-2r-2$ with locality $r$. The paper also constructs $(r,3)$-LRCs by extending every block, with optimality when the number of blocks equals the dimension parameter $t$.","feed_headline":"One extra coordinate keeps local repair optimal — length hits q+2","feed_subtitle":"Works for rational and elliptic function fields, preserving optimal minimum distance.","key_machinery":"The machinery is the function-space evaluation code together with a one-coordinate extension that records the top coefficient of the evaluated function. For rational fields, the function space $V=\\{\\sum_{i=0}^{r-1}\\sum_{j=1}^t a_{i,j}f_jz^i\\}$ has dimension $rt$, and the appended symbol $a_{r-1,t}$ is constant on the selected block, so any $r$ values of that block recover it by the same Lagrange interpolation that recovers the other symbols; the minimum-distance proof splits on whether $a_{r-1,t}$ is zero. For elliptic fields, the space $V_{E,1}$ replaces part of the basis by powers of $\\hat z$ so that the appended coordinate $a_{0,t}$ is again recoverable from the block values and the distance bound follows from the divisor $G=(t-1)(P_1+\\cdots+P_{r+1})$.","core_discovery":"The central discovery is that if $C$ is an optimal LRC whose repair blocks are each $[r+1,r]$ MDS codes, then appending one extra coordinate that records a carefully chosen coefficient, the top-degree term $a_{r-1,t}$ of the defining function, to one block yields a code that is still an optimal LRC. In the rational-function-field setting this gives optimal LRCs of length $s(r+1)+1$, dimension $rt$ and locality $r$ for the three listed conditions on $r$ and $2\\le s\\le (q+1-2r)/(r+1)$; when $r+1$ divides $q+1$, the length reaches $q+2$. The same mechanism, using the function space $V_{E,1}$ built from a rational function $\\hat z$ with $(\\hat z)=Q_1-Q_\\infty$, extends optimal elliptic LRCs to length $s(r+1)+1$ and preserves optimality for $s\\le \\lfloor(N-2r-4)/(r+1)\\rfloor$ on a curve with $N$ rational points. Extending all $s$ blocks instead produces $(r,3)$-LRCs of length $s(r+2)$, with optimality when $t=s$.","pith_inferences":["If the extension operation is applied to a different block after the first, the appended coordinate would generally break the locality argument; a testable question is whether sequential extension of $m$ distinct blocks preserves optimality for $m>1$, and for which $m$.","The unproved $(r-1)\\times(r-1)$ minor condition in the elliptic $(r,3)$ construction could be checked computationally on curves with known automorphism groups; if it fails generically, the construction may require selecting special block points.","The extension view recasts the added coordinate as a parity node attached to a repair set; in distributed-storage terms this suggests repair schemes where one extra node holds a linear function of a repair group's data.","One could test whether the rational-field length bound $q+2$ is tight for this method: the same argument does not immediately extend to $q+3$ because two appended coordinates would need to be simultaneously recoverable."],"forward_implications":["Over $\\mathbb{F}_q$, optimal LRCs with locality $r$ exist with length $q+2$ whenever $r+1$ divides $q+1$, exceeding the previous $q+1$ bound for rational-function-field constructions.","On an elliptic curve with $N$ rational points, the single-coordinate extension gives optimal LRCs of length $s(r+1)+1$ for $s\\le \\lfloor(N-2r-4)/(r+1)\\rfloor$, reaching $q+2\\sqrt{q}-2r-2$.","Extending every block yields optimal $(r,3)$-LRCs of length $s(r+2)$ and dimension $rt$ in the rational case when $t=s$, meeting the corresponding Singleton-type bound.","The Roth-Lempel type extension gives LRCs with distance at least $n-rt-t+1$, one below optimal, and reaches optimal under extra element-sum conditions.","The extension idea applies to any LRC whose repair blocks are MDS codes, not only those arising from function fields."],"supporting_citations":[{"why":"supplies the modified algebraic-geometry construction and rational-function-field LRCs of length $q+1$ that the paper extends.","marker":"[18]"},{"why":"gives the general algebraic-curve construction and the optimal LRC framework whose repair blocks are MDS codes.","marker":"[17]"},{"why":"provides the elliptic-curve LRC family and the MDS-block property (Lemma 4) that the extension preserves.","marker":"[22]"},{"why":"establishes optimal LRCs via good polynomials, the starting construction the paper cites.","marker":"[11]"},{"why":"states the Singleton-type bound for $(r,\\delta)$-LRCs used to certify optimality.","marker":"[10]"},{"why":"defines the Roth-Lempel non-Reed-Solomon MDS construction that the paper adapts to LRC extensions.","marker":"[25]"},{"why":"extends elliptic-curve automorphism groups to locality $r>23$, providing background for the elliptic section.","marker":"[19]"}],"fun_headline_variants":["Extend one coordinate, optimal LRC stays optimal, length reaches q+2","Adding a single coefficient per block yields optimal LRCs up to q+2","Optimal local repair codes can be longer: extend one block, keep optimality","From rational to elliptic: one extra coordinate preserves optimal LRC length"],"cache_read_input_tokens":20352,"weakest_assumption_plain":"For the elliptic $(r,3)$-construction, every $(r-1)\\times(r-1)$ submatrix of the block evaluation matrix $M'_u$ must be invertible, a stronger condition than the $r\\times r$ condition the paper inherits from the prior elliptic construction, and the paper neither proves it nor cites a construction that guarantees it.","fun_headline_variants_meta":{"raw":{"variants":["Extend one coordinate, optimal LRC stays optimal, length reaches q+2","Adding a single coefficient per block yields optimal LRCs up to q+2","Optimal local repair codes can be longer: extend one block, keep optimality","From rational to elliptic: one extra coordinate preserves optimal LRC length"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000354,"raw_usage":{"total_tokens":1994,"prompt_tokens":1083,"completion_tokens":911,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":699,"completion_tokens_details":{"reasoning_tokens":827}},"tokens_in":699,"tokens_out":911,"duration_ms":7902,"temperature":1.0,"reasoning_tokens":827,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T21:45:13.013840+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $q=64$ and an elliptic curve over $\\mathbb{F}_{64}$ with automorphism group of order $24$, choose $r=7$ and a block of points as in Section 2.5, and compute the $6\\times 6$ minors of the matrix $M'_u$; if any of these $(r-1)\\times(r-1)$ minors vanishes for a block that satisfies the $r\\times r$ condition of [22, Lemma 4], then the hypothesis of Theorem 13 fails and the constructed $(r,3)$-LRC is not guaranteed by the paper's proof.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"extends elliptic-curve automorphism groups to locality $r>23$, providing background for the elliptic section."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"supplies the modified algebraic-geometry construction and rational-function-field LRCs of length $q+1$ that the paper extends."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"gives the general algebraic-curve construction and the optimal LRC framework whose repair blocks are MDS codes."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"provides the elliptic-curve LRC family and the MDS-block property (Lemma 4) that the extension preserves."},{"cited_title":"I EEE Trans","cited_arxiv_id":null,"evidence_quote":"establishes optimal LRCs via good polynomials, the starting construction the paper cites."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"states the Singleton-type bound for $(r,\\delta)$-LRCs used to certify optimality."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"defines the Roth-Lempel non-Reed-Solomon MDS construction that the paper adapts to LRC extensions."}],"review_version":1}