{"id":"73a40d06-7377-45db-a5b9-6044751ba32d","arxiv_id":"2512.08100","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new family of locally recoverable codes with availability 2 is built by evaluating functions on points of a curve M_r inside the fibered surface E_r; for locality r=3 the minimum-distance lower bound is sharp.","lead":"The paper constructs error-correcting codes for distributed storage from a family of algebraic surfaces, where each data symbol can be recovered from two disjoint small sets of other symbols. The construction gives a sharp minimum-distance guarantee for the practically relevant locality parameter r=3, and is the first code construction to use a doubly elliptic K3 surface.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.4 does not prove existence of a nice nonzero t: the Chebotarev error term is dropped, and a zero place would not give the claimed evaluation grid.","rationale":"The central claim is a parameter family of LRCs with availability 2: length b(r+1)^2, dimension r(r−1)−1, locality r, availability 2, and the stated distance lower bound. Before any of these parameters can be discussed, the construction requires at least one nice \\bar t so that S_B is nonempty; this is exactly what Lemma 3.4 is supposed to guarantee. The reader's weakest assumption identifies this step, and I agree. The proof's use of Chebotarev drops the error term: Eq. (11) guarantees only that the main term is ≥2, while the O(q^{m/2}/m) error can be much larger for parameter sets satisfying the hypothesis. The r=7, q=9, m=6 illustration is concrete: the main term is about 2.2, while the error term is about 121.5, so the cited theorem cannot certify even one place. This is not a cosmetic omission. The zero-place issue compounds it: a completely splitting place at t^{r+1}=0 would give t=0, whose orbit under multiplication by ζ has one element, contradicting the partition of G_m into (r+1)-element sets and the claimed size of S_B. I do not see a comparably serious flaw in the locality/distance arguments: the Vandermonde recovery in Lemma 4.2 is sound, and the extremal-zero analysis in Proposition 4.10 is plausible, though not fully formal. Table 1 gives computational support for r=3, but only for small cases. The paper should therefore remain conditional pending a rigorous existence proof for a nonzero nice \\bar t, either via an effective Chebotarev bound or an explicit finite-field construction.","tokens_in":28898,"tokens_out":23692,"duration_ms":206674,"concrete_test":"Use the explicit error term in [22, Thm. 9.13B] (including constants) to recompute a lower bound for the number of degree-m places of F_q(t^{r+1}) that split completely in the Galois closure, for the admissible case r=7, q=9, m=6 (where Eq. (11) holds but the asymptotic error dominates). If the lower bound is not ≥2 and excludes the zero place, Lemma 3.4 is unproved for this case; then run a finite-field search over F_{9^6} for a nonzero \\bar t with P_{\\bar t} splitting into 8 distinct linear factors. Finding such a \\bar t would support the lemma's conclusion; not finding it would make the construction empty for that parameter set.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.4 is the existence step that underlies the entire construction: if no nice \\bar t exists, then S_B (Definition 3.7) is empty and C(B,L) is undefined. The proof invokes Chebotarev and writes the number of completely splitting places as q^m/(m|G|)+O(q^{m/2}/m), then drops the error term. From q^m/m ≥ 2(r+1)! only the main term ≥2 follows. For admissible parameters satisfying (11), the error term can dwarf the main term: with r=7, q=9, m=6 one has q^m/(m(r+1)!) ≈ 2.2 while q^{m/2}/m ≈ 121.5, so no positivity is obtained. The lemma may be true, but the proof as written does not establish even one evaluation point. A second gap in the same step: only the pole of t is excluded; if the completely splitting place is t^{r+1}=0, then \\bar t=0, and {ζ^j\\bar t} has size 1, not r+1, so the claimed (r+1)^2 evaluation grid cannot be built. Both issues concern nonemptiness of the code, not the distance bound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs algebraic-geometry codes from a family of fibered surfaces. For odd r ≥ 3 and q ≡ 1 mod r+1, the authors define a surface E_r and a curve M_r, choose b(r+1)^2 evaluation points obtained from completely splitting places, and evaluate the bidegree-restricted space L defined in Eq. (18). They claim the resulting codes C(B,L) have length b(r+1)^2, dimension r(r−1)−1, locality r, availability 2, and minimum distance d ≥ n − (2r^2 − 2r − 3). For r = 3 the bound is claimed to be sharp, and for b = 1 the distance is claimed to be exactly 8. Section 5 reinterprets the r = 3 case in terms of two genus-1 fibrations on a K3 surface.","tokens_in":29275,"tokens_out":19963,"duration_ms":171026,"significance":"If the central construction and the distance estimates are correct, the paper gives a new infinite family of LRCs with availability 2 from a geometric source different from previous constructions. The recovery procedure via Vandermonde systems is explicit and clean, the lower distance bound is derived directly from pole divisors rather than fitted to examples, and the r = 3 examples in Table 1 provide independent brute-force support. The interpretation via a doubly elliptic K3 surface is attractive and, if properly substantiated, would be a novel contribution. However, two load-bearing points currently lack rigorous proofs: the existence of evaluation points and the exact d = 8 statement.","major_comments":[{"comment":"The existence of a nice t-bar is not proved. The proof writes the number of completely splitting places as q^m/(m|G|) + O(q^{m/2}/m) and then drops the error term. From q^m/m ≥ 2(r+1)! one only obtains q^m/(m|G|) ≥ 2; the O-term can be much larger, e.g. r = 7, q = 9, m = 6 gives q^m/(m(r+1)!) ≈ 2.2 and q^{m/2}/m ≈ 121.5. Moreover, the argument excludes only the pole of t; if the completely splitting place is t^{r+1} = 0, then t-bar = 0 and the set H_P in Eq. (9) has size 1, so the claimed (r+1)^2 evaluation grid cannot be formed. Since Lemma 3.4 is the only nonemptiness statement for S_B, the code C(B,L) may be undefined.","section":"3, Lemma 3.4"},{"comment":"The exact-distance proof for b = 1 is incomplete. After displaying f_min in Eq. (34), the proof asserts that no function in L has more than r^2 + 2r − 7 zeros in S_t-bar, but the supporting 'maximality' argument is informal: it assumes the extremal configuration must consist of r − 1 full vertical fibers together with r − 3 additional horizontal fibers. Partial vanishing patterns are not ruled out. The claim may be true, and Table 1 supports it for r = 3, but the general odd-r case requires a rigorous extremal argument. Since d = 8 is one of the headline parameter claims, this gap is load-bearing.","section":"4.2, Proposition 4.10"},{"comment":"The alternative recovery proof rests on several assertions that are not fully justified: the statement that the trace of the image of M in the jacobian fibration is contained in the 2-torsion of the generic fiber; the application of [24, Thm 2.4]; and the conclusion from [26, Specialization thm.] that the sum of the four intersection points lies in the 2-torsion of the fiber. Because this section underpins the abstract's K3-surface claim, these statements need either precise proofs with all definitions or an explicit disclaimer that Section 5 is only a sketch. The code parameters themselves are already established by Lemma 4.2.","section":"5, Lemma 5.2"}],"minor_comments":[{"comment":"Several cross-references are to 'Theorem 2.3', 'Theorem 3.4', 'Theorem 3.7', etc., where the cited items are Definitions or Lemmas. The numbering and labels should be harmonized.","section":"Throughout"},{"comment":"The notation for q^m is ambiguous: entries like '72' and '112' are hard to read as powers. Also, some listed pairs (q,m) do not satisfy Eq. (11); the authors should clarify whether those rows are direct computational existence checks rather than consequences of Lemma 3.4.","section":"Table 1"},{"comment":"The definitions of nice point and nice t-bar should explicitly require t-bar ≠ 0. The subsequent claim that t-bar, ζ t-bar, ..., ζ^r t-bar gives (r+1)^2 distinct points depends on this; as written, the case t-bar = 0 is not excluded.","section":"Definitions 3.3 and 3.5"},{"comment":"The assertions about the singular fibers (2I_8, 4I_2) and the discriminant are stated without derivation. Since these are used to apply the Shioda-Tate formula, a short computation or a precise reference would improve readability.","section":"5, proof of Lemma 5.2"}],"recommendation":"major_revision","confidential_remarks":"I am not recommending rejection: the construction is appealing, the recovery argument in Lemma 4.2 is sound, and the two main gaps are plausibly repairable. However, acceptance should wait for a correct existence lemma with a quantitatively valid Chebotarev estimate and an explicit exclusion of t-bar = 0, and for a complete proof of the exact distance claim in Proposition 4.10."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: the paper's central existence lemma, Lemma 3.4, is not proven as written. The authors invoke Chebotarev, write the number of completely splitting places as q^m/(m|G|)+O(q^{m/2}/m), and then drop the error term. From q^m/m ≥ 2(r+1)! they get the main term is at least 2, but the error term can be much larger; for admissible parameters r=7, q=9, m=6 the main term is about 2.2 and the error about 121.5, so no place is guaranteed. They also only exclude the pole of t, not the zero place: if the splitting place is t^{r+1}=0 then \\bar t=0 and the set {ζ^j \\bar t} has size 1, so the (r+1)^2 evaluation grid collapses. Without a nonzero nice \\bar t, S_B can be empty and the code is undefined. This is a load-bearing gap.\n\nNow what is genuinely good. The family E_r/M_r is new, the vector space L (dimension r(r−1)−1, locality r, availability 2) is well chosen, and the Newton-arc pole-counting argument for the distance bound is clean. The lower bound d ≥ n − (2r^2−2r−3), the sharpness for r=3 with brute-force examples, and the exact d=8 for b=1 are real results. The dimension and recovery proofs are correct and readable. The K3 reinterpretation in Section 5 is a nice touch, and the claim about \"first code from a doubly elliptic K3 surface\" is credible.\n\nSoft spots beyond Lemma 3.4. The exact-distance proof in Prop 4.10 has an informal maximality step; I think it's right, but a referee will want it tightened. Section 5 asserts singular fiber types without showing the discriminant; minor, since Section 5 is a reinterpretation, not the main theorem. The paper honestly notes the lack of asymptotics, which is fine for this kind of construction.\n\nWho it's for: people working on AG codes from surfaces and LRC constructions. It deserves a serious referee, but only after the existence question is fixed. I would not cite it yet, and I would tell the authors to replace the Chebotarev estimate with an effective version or a direct counting argument, and to handle the t=0 case explicitly.\n\nRecommendation: send to peer review, major revision rather than desk reject. Reading group: yes, it's a good example of a fixable gap in an otherwise solid construction.","headline":"New LRC family with availability 2, but the existence of the evaluation set rests on a Chebotarev step that does not close as written.","tokens_in":29679,"tokens_out":4113,"would_cite":false,"duration_ms":36536,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H05","14G50","94B27"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fibered surfaces give locally recoverable codes with availability 2, dimension r(r−1)−1, and a sharp distance bound at r=3.","keywords":["Locally Recoverable Codes","availability","AG codes","one-variable function fields","fibered surfaces","K3 surfaces","Newton arcs","minimum distance"],"falsifier":"Check whether the set G_m of nice elements is non-empty for a triple (q, r, m) that just satisfies q^m/m ≥ 2(r+1)!, e.g., r=3, q=5, m=4, by direct search in F_{5^4} for a t such that P_t(T) = T^4 + 2T^2 − T^3 + T^2(t^4+1) − T t^4 + 1 splits into four distinct linear factors. If none exists, the construction has no evaluation points.","tokens_in":28831,"feed_emoji":"🔄","tokens_out":8821,"duration_ms":67648,"temperature":0.7,"pith_summary":"This paper asks whether algebraic surfaces that fiber over a line in two different ways can be used to build locally recoverable codes — codes in which every symbol can be recovered from a small number (r) of other symbols, and indeed from two independent such sets. The answer is yes for a specific family of surfaces E_r and curves M_r embedded in them: evaluating a carefully chosen space of polynomials on a set of 'nice' rational points gives codes with length b(r+1)^2, dimension r(r−1)−1, locality r, and availability 2. The authors prove a lower bound on the minimum distance, and show that for locality r=3 the bound is sharp; for the single-block case b=1 the distance is exactly 8. A reader interested in distributed storage would care because availability 2 means a failed node can still be repaired even if one recovery route is blocked. The construction is also the first to use a doubly elliptic K3 surface as the geometric home of an error-correcting code.","feed_headline":"Fibered surfaces yield codes with two recovery sets per symbol","feed_subtitle":"Yields LRCs with availability 2, exact distance 8 for one block, and a first-time K3 code.","key_machinery":"The machinery is the pair of fibrations on E_r: projection to t and projection to x, together with the curve M_r, a multisection of degree r+1 of both. The two fibrations supply the two recovery sets: fixing x gives a horizontal fiber, fixing t gives a vertical fiber. The function space L has degree less than r in each variable, so on each fiber the values of a function are governed by a Vandermonde system, which is what makes recovery by interpolation possible. The minimum-distance estimate is extracted from the Newton arc of the polynomial P_t(T) defining M_r at the place t=∞, which determines the valuations of elements of L at the poles; the place P_1 is a common zero of all functions in","core_discovery":"Take an odd integer r ≥ 3, a prime power q ≡ 1 (mod r+1), and a finite field F_{q^m}. On the surface E_r: y^2 = x^3 − x^2(t^{r+1}+1) + x t^{r+1}, single out the curve M_r given by y = x^{(r+1)/2}+1. Call a rational point P on M_r 'nice' if its horizontal fiber (fixed x) and vertical fiber (fixed t) each contain r further F_{q^m}-rational points of M_r. Choose b orbits of such nice points and evaluate the space L of polynomials x^i t^j with i=1,…,r−2, j=0,…,r−1 together with x^{r−1}t^h, h=0,…,r−2, on the resulting set S_B of b(r+1)^2 points. The paper proves the image is a code with dimension r(r−1)−1, locality r, availability 2, and minimum distance at least n − (2r^2 − 2r − 3); for r=3 this","pith_inferences":["A natural next step is to search, for small q and r, for 'nice' elements t of F_{q^m} when q^m/m is only slightly above 2(r+1)!; the Chebotarev bound used in the paper likely overestimates the smallest m, and a computational census could show the true threshold and whether the construction works well beyond the stated guarantee.","Because the two recovery mechanisms are independent (Vandermonde interpolation on fibers versus elliptic-curve group law on the K3 model), one could vary the space L or the curve M_r while keeping the two fibrations, obtaining availability-2 LRC families with different rate–distance tradeoffs.","The role of the common zero P_1 of all functions in L in the distance bound suggests a design principle: for a fibered-surface construction, choosing L with a large common zero outside the evaluation set may improve the lower bound; this is implicit in the paper's −2 correction."],"forward_implications":["For b=1, every code C(B,L) has minimum distance exactly 8, independent of r.","For r=3 and b≥2, the lower bound n − (2r^2 − 2r − 3) is attained in the explicit examples computed in Section 4.3.","The construction provides the first example of an error-correcting code built from a doubly elliptic K3 surface.","Each symbol of a codeword has two disjoint recovery sets of size r, so losing one recovery set does not compromise locality."],"fun_headline_variants":["First K3 surface code with twin recovery sets","Fibered surfaces make LRCs with dual repair options","Sharp distance bound for r=3 LRCs from geometry","Doubly elliptic surfaces spawn codes with two repairs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the bound in Lemma 3.4 really guarantees at least one completely splitting place, but its proof drops a Chebotarev error term O(q^{m/2}/m) whose size relative to the main term is not controlled by the stated inequality.","fun_headline_variants_meta":{"raw":{"variants":["First K3 surface code with twin recovery sets","Fibered surfaces make LRCs with dual repair options","Sharp distance bound for r=3 LRCs from geometry","Doubly elliptic surfaces spawn codes with two repairs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000385,"raw_usage":{"total_tokens":1873,"prompt_tokens":743,"completion_tokens":1130,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":1065}},"tokens_in":487,"tokens_out":1130,"duration_ms":9555,"temperature":1.0,"reasoning_tokens":1065,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T17:48:21.984749+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the set G_m of nice elements is non-empty for a triple (q, r, m) that just satisfies q^m/m ≥ 2(r+1)!, e.g., r=3, q=5, m=4, by direct search in F_{5^4} for a t such that P_t(T) = T^4 + 2T^2 − T^3 + T^2(t^4+1) − T t^4 + 1 splits into four distinct linear factors. If none exists, the construction has no evaluation points.","supporting_citations":[],"review_version":1}