{"id":"1ca5594c-6f5f-4630-9a99-9a1abcc4b813","arxiv_id":"1908.08097","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The eigenvalues of generalized Paley graphs are computed from Gaussian period sums and are shown to determine, and be determined by, the weight distributions of the associated irreducible cyclic codes when k divides (q-1)/(p-1).","lead":"This paper computes the full eigenvalue lists for several families of graphs made from finite fields, and proves that those lists are exactly determined by the weight distribution of a related family of error-correcting codes. It also identifies exactly which of these graphs are Ramanujan expanders, which are useful in communication networks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1 holds; the one concrete flaw is Theorem 2.1's complement multiplicity for disconnected Γ, which is peripheral to the central claim.","rationale":"I read the central claim as Theorem 5.1: the affine relation λ_γ = n − (p/(p−1))w(c_γ) between graph eigenvalues and code weights, with multiplicities matched when the graph is connected. I verified the relation from the Gaussian-period weight formula: for k | (q−1)/(p−1), the subgroup R_k contains F_p^*, so the number of zero coordinates of c_γ is (1/p)(n + (p−1)η_i), which yields (5.2); substituting λ = η_i gives (5.1). Connectedness makes the trace map γ ↦ c_γ injective, so the multiplicity statement in part (b) is justified, although the paper could have stated this explicitly. I found no error in the central relation itself. The external dependence on Ding–Yang and Schmidt–White formulas is real but not a defect: the paper invokes them exactly under the hypotheses needed, and the small cases I checked (Γ(3,25), Γ(3,64), Γ(4,625), and the exceptional table rows) are consistent with the stated formulas. The most serious verifiable flaw is in Theorem 2.1's complement spectrum for disconnected graphs, where the multiplicity of the complement's principal eigenvalue is mis-stated as 1+μn. This is a genuine false statement, but it is peripheral: the paper only applies complement spectra to connected semiprimitive graphs, for which μ=0 and the formula is correct. There is also the numerical typo in Table 3 for (37,7^9), where e should be 28,277 rather than 282,771; this is consistent with the reader's conditional verdict. Since neither issue touches the central claim, the reader's CONDITIONAL verdict remains appropriate.","tokens_in":24746,"tokens_out":38806,"duration_ms":370067,"concrete_test":"Run an explicit spectral computation for q=16, k=5 (e.g., in Sage): construct the Cayley graph Γ(5,16) on F_16 with connection set R_5 = {x^5 : x ∈ F_16^*}, compute Spec(Γ) and Spec(Γ̅) directly from the adjacency matrices. If Spec(Γ̅) is {[12]^1, [0]^3, [−4]^{12}} rather than {[12]^4, [0]^{12}}, then Theorem 2.1's complement multiplicity formula is false as stated. Additionally, check that the multiplicity of the largest eigenvalue of Γ̅ is 1 even though μ=1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central dictionary relation (5.1) is sound: under k | (q−1)/(p−1), F_p^* ⊆ R_k, so equation (5.2) follows by counting zeros of Tr(γx) over the relevant coset, and connectedness makes the map γ ↦ c_γ injective, which justifies identifying graph eigenvalue multiplicities with the code weight frequencies A_w in Theorem 5.1(b). The most concrete correctness defect I can verify is in Theorem 2.1, not Theorem 5.1. The stated complement spectrum {[(k−1)n]^{1+μn}, [−1−η_i]^{μ_i n}} is false when μ>0: the complement's largest eigenvalue always has multiplicity 1, while the μn eigenvectors from the Γ-principal eigenspace orthogonal to the all-ones vector become the eigenvalue −1−n. For example, take q=16 and k=5. Then Γ(5,16) ≅ 4K_4, with n=3 and μ=1. Theorem 2.1 predicts Spec(Γ̅) = {[12]^4, [0]^{12}}, but the actual complement is K_{4,4,4,4} with spectrum {[12]^1, [0]^3, [−4]^{12}}. All later uses of complement spectra occur in the connected semiprimitive setting μ=0, so Theorem 3.3 and the Ramanujan claims survive; the error does not undermine the central claim, but Theorem 2.1 as stated is false and should be corrected.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies generalized Paley graphs Γ(k,q)=Cay(F_q,R_k), where R_k is the subgroup of k-th powers in F_q^*, and the associated irreducible p-ary cyclic codes C(k,q) of length n=(q-1)/k. The main claim is a spectral dictionary: under the extra divisibility condition k | (q-1)/(p-1), the graph eigenvalue λ_γ and the Hamming weight w(c_γ) of the corresponding codeword satisfy λ_γ = n - (p/(p-1))w(c_γ), so that the spectrum of the graph and the weight distribution of the code determine each other. Building on this relation, the authors express the spectra of Γ(k,q) and its complement in terms of Gaussian periods, give explicit spectra and strongly-regular parameters for semiprimitive pairs, characterize which semiprimitive GP-graphs are Ramanujan, compute Spec(Γ(3,q)) and Spec(Γ(4,q)) from known cyclic-code weight distributions, and tabulate spectra for the first eight exceptional Schmidt-White pairs. Theorems 4.1, 6.1, and 6.3 are supported by explicit arithmetic checks, and the central relation (5.1) is derived directly from character sums and the cited weight formula rather than from the paper's own conclusions.","tokens_in":24992,"tokens_out":10035,"duration_ms":101440,"significance":"If the central relation (5.1) stands, it is a clean and useful bridge between two well-studied objects: every known weight distribution of an irreducible cyclic code gives the spectrum of a generalized Paley graph, and conversely. The paper makes this transfer explicit in several nontrivial families, including all semiprimitive graphs, the k=3 and k=4 families, and the exceptional two-weight codes. The Ramanujan characterization in Theorem 4.1 and the primitive-divisor checks in Theorems 6.1 and 6.3 are concrete and verifiable, and the paper is honest about its reliance on the external Gaussian-period formulas of Ding-Yang and the weight formulas of Schmidt-White. The dictionary relation itself is not circular and appears sound. The main weakness is a false statement about complement spectra in Theorem 2.1 for disconnected graphs; this does not affect the central dictionary relation, which is used only in connected cases, but it must be corrected.","major_comments":[{"comment":"The formula for Spec(Γ̅(k,q)) is false when μ>0. If Γ is disconnected with eigenvalue n of multiplicity 1+μn, then the all-ones vector gives the complement eigenvalue (k−1)n with multiplicity 1, while the μn principal eigenvectors orthogonal to the all-ones vector give eigenvalue −1−n, not (k−1)n. A concrete counterexample is q=16, k=5: here Γ(5,16)≅4K_4, with n=3 and μ=1, and the stated complement spectrum {[12]^4,[0]^{12}} is not the spectrum of the actual complement K_{4,4,4,4}, which is {[12]^1,[0]^3,[−4]^{12}}. The correct statement should replace the principal part of the complement spectrum with {[(k−1)n]^1, [−1−n]^{μ n}} and keep [−1−η_{i_j}]^{μ_{i_j} n} for the periods η_{i_j}≠n. Since all later uses of complement spectra occur in the connected case μ=0, the main dictionary and the subsequent theorems survive, but Theorem 2.1 as stated is incorrect.","section":"Theorem 2.1, Eq. (2.2)"},{"comment":"The theorem announces the spectra for all eleven exceptional pairs, but the proof and Tables 3–6 actually provide data only for the first eight; the pairs (163,41^81), (323,3^144), and (499,5^249) are explicitly omitted as 'quite unmanageable'. Either the statement should be restricted to the eight pairs for which spectra are given, or the missing spectra should be supplied, so that the theorem's claim matches the evidence presented.","section":"Section 7, Theorem 7.1"}],"minor_comments":[{"comment":"The proof refers to 'Proposition 2.1' when citing the spectrum of Γ(k,q); this is Theorem 2.1.","section":"Section 5, proof of Theorem 5.1"},{"comment":"The typesetting of expressions such as '−√q+1/k' is ambiguous; it should be printed as (−√q+1)/k to prevent misreading as −√q + 1/k.","section":"Equations (3.4)–(3.5)"},{"comment":"There are several typographical errors, including 'whit', 'integerdivide', and 'Propisition'; these should be corrected in a final pass.","section":"Section 1 and throughout"},{"comment":"The remark correctly notes that for k=3t and k=4t with t>1, the method of Section 5 does not directly apply; this limitation should also be reflected in the introductory summary to avoid overstating the scope of the k=3 and k=4 computations.","section":"Remark 6.5(ii)"}],"recommendation":"major_revision","confidential_remarks":"The central dictionary relation is sound, and the false complement-spectrum statement in Theorem 2.1 is local and fixable; I do not see a reason to reject. The incompleteness of Theorem 7.1 for three exceptional pairs should be resolved either by computing them or by restricting the theorem. The paper's reliance on Ding-Yang and Schmidt-White formulas is legitimate external dependence, not circularity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a solid, useful piece: it proves a clean dictionary relation (Theorem 5.1) between the eigenvalues of a generalized Paley graph and the weights of the associated irreducible cyclic code, and it uses that to give the first complete Ramanujan classification of semiprimitive GP-graphs (Theorem 4.1), plus explicit spectra for Gamma(3,q), Gamma(4,q), and the exceptional pairs. The main arithmetic—primitive divisor checks, Ramanujan inequalities, Gaussian-period evaluations—holds up. Theorem 5.1 itself is sound: I checked the counting behind (5.1), and the connectedness assumption for multiplicities is justified.\n\nThe soft spots are real but mostly peripheral. Theorem 2.1 as stated is false for disconnected Gamma. The complement spectrum's largest eigenvalue has multiplicity 1, not 1+mu*n, and the mu*n vectors in the principal eigenspace orthogonal to the all-ones become eigenvalue -1-n. Example: q=16, k=5 gives Gamma congruent 4K_4, complement K_{4,4,4,4}, spectrum {[12]^1, [0]^3, [-4]^12}, not what (2.2) predicts. Since all later applications are in the connected semiprimitive case (mu=0), the central results survive, but the theorem needs a corrected statement. There are also smaller blemishes: the \"strongly regular\" claims at the ends of Theorems 6.1 and 6.3 are only true in the semiprimitive cases (b); case (a) has four or five distinct eigenvalues. The srg parameter e for (37,7^9) in Table 3 is inconsistent with lambda1+lambda2 and d—looks like 28,277, not 282,771. And several \"straightforward calculations\" are genuinely left to the reader, which slows verification.\n\nNone of this blocks the main contribution. The paper inherits Gaussian-period evaluations from Ding–Yang and Schmidt–White without re-deriving them, which is a caveat but not a flaw. The citation pattern is fine; the authors cite their own prior work where appropriate and clearly mark overlap in Remark 3.4.\n\nWho should read it: anyone working on generalized Paley graphs, integral strongly regular graphs, or weight distributions of irreducible cyclic codes. It deserves a serious referee; with minor revision (Theorem 2.1 correction, typos, and clarifying the srg statements) it would be a solid published paper.","headline":"A useful dictionary between GP-graph spectra and code weights, with a real but peripheral flaw in the complement formula for disconnected graphs.","tokens_in":25602,"tokens_out":3258,"would_cite":true,"duration_ms":30500,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B15","05C25","05C50","11P05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Generalized Paley graph spectra and cyclic code weights are the same datum: each eigenvalue is an affine function of a codeword weight.","keywords":["generalized Paley graphs","irreducible cyclic codes","Gaussian periods","strongly regular graphs","Ramanujan graphs","two-weight codes","Cayley graph spectra","finite fields"],"falsifier":"Take $(k,q)=(3,25)$, so $p=5$ and $k$ divides $(q-1)/(p-1)=6$; enumerate $\\mathbb{F}_{25}$, build the connection set $R_3$, form the $25\\times 25$ adjacency matrix of $\\Gamma(3,25)$, and list all 25 codewords $\\mathrm{Tr}_{25/5}(\\gamma\\omega^{3i})$ for $i=0,\\ldots,7$. If the 25 eigenvalues are not $\\{[8]^1,[3]^8,[-2]^{16}\\}$ or if $\\lambda_\\gamma = 8 - \\frac{5}{4} w(c_\\gamma)$ fails for a single $\\gamma$, the central relation is false; the same brute-force check can be run on any small pair satisfying $k \\mid (q-1)/(p-1)$.","tokens_in":24481,"feed_emoji":"📐","tokens_out":11033,"duration_ms":184634,"temperature":0.7,"pith_summary":"This paper builds a bridge between two objects that live on the same finite field: the generalized Paley graph $\\Gamma(k,q)$, whose vertices are field elements and whose edges connect elements differing by a nonzero $k$-th power, and the irreducible $p$-ary cyclic code $\\mathcal{C}(k,q)$ of length $n=(q-1)/k$. The central result is an affine identity: when $k$ divides $(q-1)/(p-1)$, the eigenvalue $\\lambda_\\gamma$ of the graph indexed by $\\gamma$ equals $n - \\frac{p}{p-1} w(c_\\gamma)$, where $w(c_\\gamma)$ is the Hamming weight of the codeword labelled by $\\gamma$. When the graph is connected, the multiplicities also match, so the full spectrum of either object determines the other. The authors use this to give explicit integral spectra in the semiprimitive case, for all $\\Gamma(3,q)$ and $\\Gamma(4,q)$ under natural divisibility assumptions, and for the eleven exceptional two-weight code pairs.","feed_headline":"A simple formula unites Paley graph spectra and code weights","feed_subtitle":"When k divides (q−1)/(p−1), each graph eigenvalue is n − p/(p−1) times a code weight, so the two spectra determine each other.","key_machinery":"The load-bearing object is the Gaussian period $\\eta_i^{(k,q)} = \\sum_{x\\in C_i} \\zeta_p^{\\mathrm{Tr}_{q/p}(x)}$, a character sum over one coset of the subgroup of $k$-th powers in $\\mathbb{F}_q^*$. For a Cayley graph over an abelian group, each eigenvalue is a character sum over the connection set; here that sum reindexes to a Gaussian period, giving $\\mathrm{Spec}(\\Gamma(k,q))$ directly from the periods. The matching formula $w(c_\\gamma)=\\frac{p-1}{pk}(q-1-k\\eta_i)$ expresses every codeword weight as an affine function of the same period. Comparing the two expressions yields the identity in Theorem 5.1 and transfers every known code-spectrum computation into a graph-spectrum computation.","core_discovery":"For $q=p^m$ and $k$ dividing $q-1$, the paper considers $\\Gamma(k,q)=\\mathrm{Cay}(\\mathbb{F}_q,R_k)$ with $R_k=\\{x^k: x\\neq 0\\}$, and the code $\\mathcal{C}(k,q)$ whose codewords are the trace vectors $c_\\gamma=(\\mathrm{Tr}_{q/p}(\\gamma\\omega^{ki}))_{i=0}^{n-1}$. The central claim, Theorem 5.1, is that if $k$ also divides $(q-1)/(p-1)$, then every eigenvalue $\\lambda_\\gamma$ of the graph, computed through additive characters of $\\mathbb{F}_q$, and the weight $w(c_\\gamma)$ of the corresponding codeword satisfy $\\lambda_\\gamma = n - \\frac{p}{p-1}w(c_\\gamma)$, with $n=(q-1)/k$. For a connected graph the multiplicity of each eigenvalue is exactly the number of codewords of the corresponding weight, so the spectrum of the graph and the weight distribution of the code determine each other completely. The paper applies this dictionary to produce explicit spectra of semiprimitive generalized Paley graphs, of $\\Gamma(3,q)$ and $\\Gamma(4,q)$ when the divisibility condition holds, and of the graphs attached to the eleven exceptional pairs; in these cases the graphs turn out to be integral and, in the two-weight cases, strongly regular.","pith_inferences":["Editorial inference: because the identity runs in both directions, any newly discovered two-weight irreducible cyclic code satisfying the divisibility condition would immediately produce a strongly regular graph, so the graph side gives an independent test of the conjecture that the exceptional list (7.1) is complete.","Editorial inference: the Ramanujan-to-distance inequality in Corollary 5.4 can be read as a coding-theoretic reflection of the optimal spectral-expansion bound for regular graphs; optimizing the same relation may yield minimum-distance bounds even for non-Ramanujan graphs in the family.","Editorial inference: replacing the absolute trace by another linear functional, or dropping the divisibility assumption, would replace the exact identity by a comparison between a twisted Cayley graph and a shortened or punctured code; this is a testable way to measure how much of the dictionary survives.","Editorial inference: the same Gaussian-period machinery gives spectra for the complementary graph and for the coset graphs in the disjoint-union decomposition of the complement, so the relation plausibly extends to those components as well."],"forward_implications":["Known weight distributions of irreducible cyclic codes automatically give spectra of generalized Paley graphs, and conversely, whenever $k$ divides $(q-1)/(p-1)$; the two computations become interchangeable.","Semiprimitive pairs $(k,q)$ give connected integral strongly regular graphs with three explicit eigenvalues, and their complements are also strongly regular; for $s$ odd they are Latin square graphs, providing complete sets of mutually orthogonal Latin squares of order $p^{m/2}$.","The only semiprimitive generalized Paley graphs that are Ramanujan are the classical Paley graphs plus the $k=3,4,5$ families listed in Theorem 4.1, and every semiprimitive complement is Ramanujan.","If the graph attached to a code is Ramanujan and the code's minimum distance $d$ satisfies $d \\le (p-1)n/p$, then $d \\ge \\frac{p-1}{p}(n - 2\\sqrt{n-1})$; this converts an expansion property of the graph into a distance guarantee of the code.","The eleven exceptional two-weight irreducible cyclic codes yield connected strongly regular graphs whose spectra and parameters are explicitly computed for the first eight pairs; the remaining three pairs are stated to be too large for readable tables."],"supporting_citations":[{"why":"It supplies the Gaussian-period evaluations and the spectra of $\\mathcal{C}(3,q)$ and $\\mathcal{C}(4,q)$ that Theorem 5.1 converts into the graph spectra in Theorems 6.1 and 6.3.","marker":"[12]"},{"why":"It supplies the exceptional-pair list (7.1) and the weight formula (7.2) from which Theorem 7.1 computes the spectra of the associated graphs and codes.","marker":"[23]"},{"why":"It supplies the semiprimitive Gaussian-period and eigenvalue analysis underlying the explicit spectra in Theorem 3.3.","marker":"[3]"},{"why":"It establishes the dictionary between two-weight codes and strongly regular graphs that motivates Corollary 5.3 and the computation of the strongly regular parameters.","marker":"[8]"},{"why":"It gives earlier spectrum computations for the subfamily $\\Gamma(p^\\ell+1,p^m)$, against which Theorem 3.3 is checked and on which Example 5.5 draws.","marker":"[19]"},{"why":"It provides weight distributions of irreducible cyclic codes with small $N$ that feed the explicit spectra for $k=3$ and $k=4$.","marker":"[10]"}],"fun_headline_variants":["Paley spectra and code weights: a simple linear link","New formula ties Paley graph eigenvalues to code weights","Eigenvalues of Paley graphs read off from code weights","When spectra of Paley graphs and cyclic codes coincide","A dictionary between Paley graph spectra and code weights"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The explicit spectra all rest on imported Gaussian-period evaluations and exceptional weight formulas that the paper does not re-derive; if those formulas carry hidden hypotheses or errors, the corresponding spectra in Theorems 3.3, 6.1, 6.3, and 7.1 inherit the failure.","fun_headline_variants_meta":{"raw":{"variants":["Paley spectra and code weights: a simple linear link","New formula ties Paley graph eigenvalues to code weights","Eigenvalues of Paley graphs read off from code weights","When spectra of Paley graphs and cyclic codes coincide","A dictionary between Paley graph spectra and code weights"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000281,"raw_usage":{"total_tokens":1732,"prompt_tokens":1080,"completion_tokens":652,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":696,"completion_tokens_details":{"reasoning_tokens":573}},"tokens_in":696,"tokens_out":652,"duration_ms":533755,"temperature":1.0,"reasoning_tokens":573,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:51:03.776736+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $(k,q)=(3,25)$, so $p=5$ and $k$ divides $(q-1)/(p-1)=6$; enumerate $\\mathbb{F}_{25}$, build the connection set $R_3$, form the $25\\times 25$ adjacency matrix of $\\Gamma(3,25)$, and list all 25 codewords $\\mathrm{Tr}_{25/5}(\\gamma\\omega^{3i})$ for $i=0,\\ldots,7$. If the 25 eigenvalues are not $\\{[8]^1,[3]^8,[-2]^{16}\\}$ or if $\\lambda_\\gamma = 8 - \\frac{5}{4} w(c_\\gamma)$ fails for a single $\\gamma$, the central relation is false; the same brute-force check can be run on any small pair satisfying $k \\mid (q-1)/(p-1)$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the Gaussian-period evaluations and the spectra of $\\mathcal{C}(3,q)$ and $\\mathcal{C}(4,q)$ that Theorem 5.1 converts into the graph spectra in Theorems 6.1 and 6.3."},{"cited_title":"Schmidt, C","cited_arxiv_id":null,"evidence_quote":"It supplies the exceptional-pair list (7.1) and the weight formula (7.2) from which Theorem 7.1 computes the spectra of the associated graphs and codes."},{"cited_title":"Baumer t, R.J","cited_arxiv_id":null,"evidence_quote":"It supplies the semiprimitive Gaussian-period and eigenvalue analysis underlying the explicit spectra in Theorem 3.3."},{"cited_title":"Delsar te","cited_arxiv_id":null,"evidence_quote":"It establishes the dictionary between two-weight codes and strongly regular graphs that motivates Corollary 5.3 and the computation of the strongly regular parameters."},{"cited_title":"The spectra of generalized Paley graphs of $(q^\\ell+1)$-th powers and applications","cited_arxiv_id":"1812.03332","evidence_quote":"It gives earlier spectrum computations for the subfamily $\\Gamma(p^\\ell+1,p^m)$, against which Theorem 3.3 is checked and on which Example 5.5 draws."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides weight distributions of irreducible cyclic codes with small $N$ that feed the explicit spectra for $k=3$ and $k=4$."}],"review_version":1}