{"id":"7bf20480-a34e-4eec-9105-d99f8071a47e","arxiv_id":"1908.01471","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims an asymptotic rate-distance tradeoff for locally repairable codes over arbitrary finite fields and arbitrary locality using local expansions at a rational place, but the construction has a critical flaw.","lead":"This paper proposes explicit error-correcting codes for distributed storage, called locally repairable codes, built from function fields, and claims they beat prior asymptotic bounds for any alphabet size and any locality parameter. A key step in the construction, however, relies on a false assumption about a matrix, so the proof as written does not establish the claim.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'without loss of generality' claim that the first g rows of A have rank g is false: the constant function lies in the kernel of the first-g-coefficient projection, so the cancellation step producing (17) is not guaranteed.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing flaw. Lemma 3.1 and even the remark that the first 2g rows have full rank are plausible, but the transition to the first g rows is not a WLOG: the rank of A1 is basis-independent and bounded by g-1 because the constants lie in L((g-1)P_infty) and hence in the kernel of the coefficient projection. Consequently the key functions f_ij with local expansion (17) are not guaranteed, and the minimum-distance proof in Proposition 3.3 depends on precisely that expansion. Section 4 copies the same step, so the later theorems inherit the gap. No machine-checked verification or alternative argument in the text repairs this. I confirm the reader's REJECT verdict; no adjustment is needed.","tokens_in":14239,"tokens_out":14207,"duration_ms":149356,"concrete_test":"Take a genus-2 hyperelliptic function field over F_q with a rational place P_infty and pole numbers 0,2; let {1,x} be a basis of L(3P_infty). Compute the local expansions at P_infty and form A1 from the first g=2 rows. The column for 1 is (0,0), so rank(A1)<2. Choose any g_ij extending this basis to L(3P_infty+P_ij) whose expansion has b_0 nonzero, and run Gaussian elimination on A1 x = (b_0,b_1); it is unsolvable. More generally, prove that L((g-1)P_infty) is contained in the kernel of the first-g-coefficient projection and that it contains 1, so rank(A1)<=g-1 always.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 needs f_ij = g_ij - sum_w alpha_w f_w to have a local expansion at P_infty starting at p=g (eq. (17)). This is obtained by solving A1 x^T = (b_0,...,b_{g-1})^T, where A1 is the first g rows of the matrix A in (15). The paper says 'Without loss of generality' that A1 has rank g. This is impossible. For every f in L((g-1)P_infty), the pole order at P_infty is at most g-1, so its expansion pi^{-2g+1} sum_i c_i pi^i has c_0=...=c_{g-1}=0. Since L((g-1)P_infty) always contains the constant function 1, it is a nonzero subspace of the kernel of the projection L((2g-1)P_infty) to F_q^g given by the first g coefficients. Hence that projection has rank at most g-1 for every choice of basis, so no basis makes A1 have rank g. The linear system A1 x^T = b therefore need not be solvable for arbitrary g_ij, and the existence of f_ij with the claimed expansion is not established. Proposition 3.3 later uses (17) to assert that a linear combination of selected columns has local expansion pi^{-2g+1} sum_{p=2g+t} ..., and the distance bound d>t+1 depends on that cancellation. Without (17), the degree argument collapses. Section 4 says 'in the same way as in Section 3' and inherits the same gap, so Theorem 1.4 and Corollary 1.2 do not follow as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an explicit asymptotic construction of q-ary linear locally repairable codes with arbitrary locality r, using local expansions of functions from Riemann-Roch spaces at a rational place P∞. The main result, Theorem 1.1, claims a Tsfasman-Vladut-Zink type lower bound R > r/(r+1) - (r/(r+1))/A(q) - δ for any prime power q and any locality r, with explicit versions for square and odd-power prime fields in Corollary 1.2, and a separate construction over prime fields in Theorem 1.4. The construction forms a parity-check matrix whose columns are coefficients of local expansions of functions f_ij, obtained by subtracting from each g_ij a linear combination of a basis of L((2g-1)P∞) to eliminate the first g local-expansion coefficients. The claimed distance bound depends on this elimination step.","tokens_in":14565,"tokens_out":8477,"duration_ms":82305,"significance":"If the main theorem were correct, the paper would be a substantial contribution: it would give locally repairable codes with no restrictions on alphabet size or locality, improve on earlier automorphism-group constructions for square alphabets, and exceed the Gilbert-Varshamov type bound for non-prime fields with large alphabet in some regimes. The paper is self-contained, uses standard function-field theory, and makes explicit comparisons with prior bounds. However, the central algebraic step on which the entire construction rests is invalid. Because the claimed rank of the first g rows of the local-expansion matrix is impossible, the functions f_ij with the required vanishing coefficients are not guaranteed to exist, and the distance proof collapses. The significance is therefore conditional, and the results as stated are not established.","major_comments":[{"comment":"The 'Without loss of generality' statement that the submatrix A1 consisting of the first g rows of A has rank g is false. For every basis {f_1,...,f_g} of L((2g-1)P∞), the constant function 1 belongs to this space, and in its expansion π^{-2g+1}∑ c_i π^i the first nonzero coefficient appears at i = 2g-1. Hence c_0 = ... = c_{g-1} = 0, so the constant function lies in the kernel of the projection L((2g-1)P∞) → F_q^g given by the first g expansion coefficients. This projection therefore has rank at most g-1 for every choice of basis, so no basis can make A1 have rank g. Consequently the linear system A1 x^T = (b_0,...,b_{g-1})^T need not be solvable for the chosen g_ij, and the existence of f_ij with the claimed local expansion (17) is not guaranteed. Remark 1(i) only proves that the first 2g rows have rank g, which is compatible with this obstruction.","section":"Section 3, eq. (15) and the paragraph before eq. (16)"},{"comment":"The distance proof in Proposition 3.3 relies on the step asserting that the local expansion of ∑_{i∈I}∑_{j∈S_i} λ_ij f_ij is π^{-2g+1}∑_{p=2g+t} ... π^p. This implication uses (17) and the fact that D_i records only coefficients a_{p,ij} for p ≥ g. Without (17), the selected combination may have nonzero coefficients a_g, ..., a_{2g-1+t}, so the valuation at P∞ need not exceed 1+t. The subsequent degree argument involving deg(−(1+t)P∞ + ∑∑ P_ij) = -1 then cannot be applied, and the claim d > t+1 is unsupported. Since Theorem 1.1 is derived directly from Proposition 3.3, the main asymptotic result does not follow as written. Section 4 constructs f_ij 'in the same way as in Section 3' and therefore inherits the same gap, so Theorem 1.4, Corollary 1.2, and the claimed GV-type improvements in Proposition 1.3 and the figures are also not justified.","section":"Proposition 3.3 and Theorem 1.1"}],"minor_comments":[{"comment":"The text says 'For 1 ⩽ j⩽ n, the local expansions of f_j ...' but the basis has g elements; this should read '1 ⩽ j⩽ g'.","section":"Section 3, first paragraph after eq. (14)"},{"comment":"In both cases the notation 'S_u \\ {1, i+1}' should be 'S_u \\ {1, r+1}', since f_{u,r+1} = α_u f_{u,1} is the repeated column.","section":"Proposition 3.3, proof, Cases 1 and 2"},{"comment":"The word 'Riemman-Roch' is misspelled; it should be 'Riemann-Roch'.","section":"Propositions 3.3 and 4.2"},{"comment":"The line 'Let m_i b + 1 = ∑_{d|e} d·B_d(F_i)' is unclear: since the sum equals N(E_i), it appears to define m_i, but the notation should be made explicit, and the choice of disjoint effective divisors of degree b requires justification.","section":"Proof of Theorem 1.4"}],"recommendation":"reject","confidential_remarks":"The central flaw is decisive: the claimed rank of the first g rows of the local-expansion matrix is impossible because constants (and more generally L((g-1)P∞)) lie in the kernel of the first-g-coefficient projection. This is not a local fix; the construction of f_ij with vanishing first g coefficients is not possible in general. If the authors can supply a genuinely different method to obtain such functions, the paper would need to be substantially rewritten and re-reviewed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe paper has a genuinely new idea: instead of using automorphism groups and divisibility conditions on r+1, the authors construct locally repairable codes from local expansions at a rational place, aiming for a TVZ-type bound with no restrictions on alphabet size or locality. That would be a substantial step if it worked. The exposition is clear, the full rank lemma for the matrix A is correct, and the general parity-check framework is plausible. The references and comparisons with prior bounds look fair.\n\nThe central step in Section 3 is wrong. The authors state \"Without loss of generality\" that the first g rows of A have rank g. This cannot happen. For every basis of L((2g-1)P∞), the constant function 1 has local expansion with zero coefficients in rows 0 through g-1; its first nonzero coefficient appears at row 2g-1. Equivalently, all of L((g-1)P∞) lies in the kernel of the projection to the first g coefficient rows, and 1 is such a function. So the rank of that submatrix is at most g-1, no matter what basis is chosen. The system A1 x^T = (b_0,...,b_{g-1})^T is not always solvable, and the existence of f_ij with local expansion starting at p=g (equation (17)) is not guaranteed. The distance proof in Proposition 3.3 depends directly on that cancellation; without it the degree argument collapses. Section 4 says \"in the same way as in Section 3\" and inherits the gap. Theorems 1.1 and 1.4 are not established as written.\n\nThere is also a smaller gap: the proof of Proposition 1.3 only treats δ=1/2, while the statement claims the GV bound is beaten for sufficiently large q without specifying the range of δ. That would be easy to fix if the main construction held.\n\nI agree with the reader's judgment. The stress-test note is correct: this is not a harmless normalization; it is a load-bearing false statement. That said, this is not a paper to desk-reject. The local-expansion approach is novel and might be repairable—for instance, by cancelling a different set of coefficient rows or working in a larger Riemann-Roch space. A serious referee should engage with the construction and see whether a modified argument can restore the bound. As written, the main claims do not follow.\n\nWho is this for? Coding theorists working on asymptotic LRC constructions and AG codes. It deserves a serious referee, but the referee should focus on the rank claim and the cancellation step.","headline":"A novel local-expansion route to asymptotic LRCs, but a false rank assumption at the center leaves the main bounds unproven.","tokens_in":15108,"tokens_out":8721,"would_cite":false,"duration_ms":78165,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B27","94B65","11G20","14H05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every prime power q and every locality r, this paper constructs linear locally repairable codes meeting a TVZ-type asymptotic rate–distance bound, with no restriction on alphabet or locality.","keywords":["locally repairable codes","asymptotic bounds","function fields","local expansions","Ihara's constant","Tsfasman–Vladut–Zink bound","Gilbert–Varshamov bound","finite fields"],"falsifier":"Take a basis of $L((2g-1)P_\\infty)$ that contains the constant function 1; since 1 has a zero in every row of $A$ before row $2g-1$, the $g\\times g$ submatrix $A_1$ formed by rows $0,\\dots,g-1$ has a zero column and rank at most $g-1$. This contradicts the assumed full-rank step and shows the linear system $A_1 x^T = (b_0,\\dots,b_{g-1})^T$ that defines the adjusted functions is not always solvable.","tokens_in":14008,"feed_emoji":"💾","tokens_out":17051,"duration_ms":153677,"temperature":0.7,"pith_summary":"Locally repairable codes are storage codes in which each erased coordinate can be recovered from at most r others, and this paper studies the best possible trade-off between code rate and relative distance as the block length grows. The paper gives an explicit construction, valid over every finite field and for every fixed locality r, that realizes a Tsfasman–Vladut–Zink-type asymptotic bound: the rate R and relative distance δ satisfy $R > \\frac{r}{r+1} - \\frac{r}{r+1}\\frac{1}{A(q)} - \\delta$, where $A(q)$ is the maximal asymptotic ratio of rational places to genus over $\\mathbb{F}_q$. Earlier curve-based constructions required the alphabet size to be a square and imposed divisibility conditions on r; the claimed advantage here is that both restrictions disappear. Over non-prime finite fields with sufficiently large alphabet, the resulting bound is shown to exceed the asymptotic Gilbert–Varshamov bound for locally repairable codes, and a separate variant over prime fields gives an explicit but weaker bound.","feed_headline":"Storage-repair codes get a curve-based rate bound for any locality","feed_subtitle":"It removes alphabet and locality restrictions, beating the Gilbert–Varshamov bound for large non-prime fields.","key_machinery":"The central object is the local expansion matrix $A$ (equation (15)): for a basis of $L((2g-1)P_\\infty)$ with pole numbers $n_j$, the columns record the coefficients $c_{i,j}$ in the local expansions $f_j = \\pi^{-2g+1}\\sum_{i\\ge 0} c_{i,j}\\pi^i$ at a rational place $P_\\infty$. The construction hinges on the 'without loss of generality' claim that the $g\\times g$ submatrix $A_1$ of the first $g$ rows has rank $g$; this lets one solve $A_1 x^T = (b_0,\\dots,b_{g-1})^T$ for each auxiliary function $g_{ij}$, so that the adjusted functions $f_{ij} = g_{ij} - \\sum_w \\alpha_{w,ij} f_w$ have expansions starting at $\\pi^g$. The blocks $D_i$ of coefficients $a_{p,ij}$ ($p = g,\\dots,2g-1+t$) form the lower part of the parity-check matrix $H$; the all-one and identity blocks at the top ensure that any single erased coordinate can be recovered from the other r coordinates in its block, i.e. locality r. Taking a tower of function fields with $N(F_i)/g(F_i) \\to A(q)$ and letting t grow proportionally to n converts the rank condition on $H$ into the asymptotic rate–distance trade-off.","core_discovery":"The paper's central discovery is that the local expansions of functions at a single rational place can be turned directly into parity-check matrices of locally repairable codes. Starting with an $\\mathbb{F}_q$-basis $\\{f_1,\\dots,f_g\\}$ of the Riemann–Roch space $L((2g-1)P_\\infty)$, each basis element has a local expansion $f_j = \\pi^{-2g+1}\\sum_{i\\ge 0} c_{i,j}\\pi^i$ at a rational place $P_\\infty$. The paper labels the basis so that the $g\\times g$ matrix $A_1$ of the first $g$ coefficients $c_{i,j}$ has full rank; for each auxiliary function $g_{ij}$ with a simple pole at a chosen repair position $P_{ij}$, it then solves $A_1 x^T = (b_0,\\dots,b_{g-1})^T$ and forms $f_{ij} = g_{ij} - \\sum_w \\alpha_{w,ij} f_w$, so that the local expansion of $f_{ij}$ starts at $\\pi^g$. The coefficients $a_{p,ij}$ for $p=g,\\dots,2g-1+t$ fill the lower blocks of a parity-check matrix whose upper blocks force locality r. Proving that any t columns of this matrix are independent gives $d > t+1$, and passing to a tower of function fields with $N(F_i)/g(F_i) \\to A(q)$ yields the asymptotic inequality of Theorem 1.1. For square alphabets the tower can be taken explicit; for $q = p^{2m+1}$ the paper uses the explicit towers of [4]; for prime alphabets, places of higher degree replace the rational repair places and produce the bound of Theorem 1.4.","pith_inferences":["Editorial inference: the parity-check template with the same structure of one identity block and one local-expansion block per repair group should also produce codes with availability, by adding several independently scaled copies of a block column, at the price of a lower rate; the paper does not pursue this.","Editorial inference: since the construction depends only on the ratio of rational places to genus and not on the automorphism group, any explicit tower of function fields with a good limit $A(q)$ should plug into Theorem 1.1, so improvements in explicit towers would automatically improve the claimed rate bound.","Editorial inference: a natural testable extension is to enforce higher-order zeros at the other repair positions, which would trade rate for larger minimum distance or additional locality properties; the paper leaves this trade-off open."],"forward_implications":["For q a square, the construction gives explicit q-ary locally repairable codes with arbitrary locality r and $R > \\frac{r}{r+1} - \\frac{r}{r+1}\\frac{1}{\\sqrt{q}-1} - \\delta$.","For $q = p^{2m+1}$, it gives explicit codes with $R > \\frac{r}{r+1} - \\frac{1}{2}\\left(\\frac{1}{p^m-1} + \\frac{1}{p^{m+1}-1}\\right)\\frac{r}{r+1} - \\delta$.","For non-prime fields with q large and $r \\ge c \\log_2 q$ for any $c > 1$, the new bound exceeds the asymptotic Gilbert–Varshamov bound for locally repairable codes.","For prime q, the same local-expansion method applied to places of high degree yields an explicit family with $R > \\frac{r}{r+1} - \\frac{b}{r+1}\\frac{1}{q^{e/2}-1} - e\\delta$, for $b = r$, $r+1$, or $r+2$ according to parity, although the paper notes this bound does not beat the Gilbert–Varshamov-type bound."],"supporting_citations":[{"why":"Earlier construction of asymptotically good locally repairable codes on algebraic curves that this paper extends and compares against.","marker":"[3]"},{"why":"Supplies explicit towers of function fields over non-prime fields used for the $q=p^{2m+1}$ case.","marker":"[4]"},{"why":"Provides the explicit optimal tower over square alphabets used to get $A(q)=\\sqrt{q}-1$.","marker":"[8]"},{"why":"Gives the genus and rational-place counts of the tower used in the construction.","marker":"[9]"},{"why":"A more flexible automorphism-group construction of asymptotically good LRCs used as a comparison bound.","marker":"[16]"},{"why":"Standard facts on Riemann–Roch spaces, local expansions, Weierstrass gaps, and constant field extensions.","marker":"[22]"},{"why":"Asymptotic Gilbert–Varshamov bound for locally repairable codes that Proposition 1.3 claims to beat.","marker":"[24]"},{"why":"Origin of the TVZ bound for classical algebraic-geometry codes that Theorem 1.1 transfers to locally repairable codes.","marker":"[26]"},{"why":"The local-expansion technique for constructing algebraic-geometry codes that the parity-check construction is based on.","marker":"[27]"}],"fun_headline_variants":["LRCs from function fields: no locality or alphabet limits","Curve-based LRCs beat GV bound for large non-prime fields","TVZ-type bound for locally repairable codes via function fields","Explicit LRC construction from rational place expansions","LRCs get TVZ bound with no alphabet or locality constraints"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything hangs on the 'without loss of generality' assumption that the first $g$ expansion coefficients of the chosen basis functions are linearly independent, so the linear system used to adjust each auxiliary function has a solution; if that system is singular, the functions carrying both locality and distance are not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["LRCs from function fields: no locality or alphabet limits","Curve-based LRCs beat GV bound for large non-prime fields","TVZ-type bound for locally repairable codes via function fields","Explicit LRC construction from rational place expansions","LRCs get TVZ bound with no alphabet or locality constraints"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001539,"raw_usage":{"total_tokens":6210,"prompt_tokens":1049,"completion_tokens":5161,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":665,"completion_tokens_details":{"reasoning_tokens":5073}},"tokens_in":665,"tokens_out":5161,"duration_ms":36183,"temperature":1.0,"reasoning_tokens":5073,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:16:21.963563+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a basis of $L((2g-1)P_\\infty)$ that contains the constant function 1; since 1 has a zero in every row of $A$ before row $2g-1$, the $g\\times g$ submatrix $A_1$ formed by rows $0,\\dots,g-1$ has a zero column and rank at most $g-1$. This contradicts the assumed full-rank step and shows the linear system $A_1 x^T = (b_0,\\dots,b_{g-1})^T$ that defines the adjusted functions is not always solvable.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier construction of asymptotically good locally repairable codes on algebraic curves that this paper extends and compares against."},{"cited_title":"Bassa, P","cited_arxiv_id":null,"evidence_quote":"Supplies explicit towers of function fields over non-prime fields used for the $q=p^{2m+1}$ case."},{"cited_title":"Garcia and H","cited_arxiv_id":null,"evidence_quote":"Provides the explicit optimal tower over square alphabets used to get $A(q)=\\sqrt{q}-1$."},{"cited_title":"Garcia and H","cited_arxiv_id":null,"evidence_quote":"Gives the genus and rational-place counts of the tower used in the construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"A more flexible automorphism-group construction of asymptotically good LRCs used as a comparison bound."},{"cited_title":"Stichtenoth, Algebraic Function Fields and Codes , Graduate Texts in Mathematics 254, Springer Verlag, 2009","cited_arxiv_id":null,"evidence_quote":"Standard facts on Riemann–Roch spaces, local expansions, Weierstrass gaps, and constant field extensions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Asymptotic Gilbert–Varshamov bound for locally repairable codes that Proposition 1.3 claims to beat."},{"cited_title":"Tsfasman, S.G","cited_arxiv_id":null,"evidence_quote":"Origin of the TVZ bound for classical algebraic-geometry codes that Theorem 1.1 transfers to locally repairable codes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The local-expansion technique for constructing algebraic-geometry codes that the parity-check construction is based on."}],"review_version":1}