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REVIEW 3 major objections 4 minor 22 references

New Nonexistence Results on Circulant Weighing Matrices

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Twelve parameter pairs long open in the small-order table for circulant weighing matrices are proven to admit no such matrix.

desk verdict Solid incremental progress on Strassler's table; the hand proofs are the real contribution, but the computer-eliminated cases need code and logs before the headline twelve nonexistence claims are fully auditable. read the letter →

arxiv 1908.08447 v3 pith:WG2QBMDZ submitted 2019-08-22 math.CO

K.T. Arasu , Daniel M. Gordon , Yiran Zhang This is my paper · ORCID
classification math.CO MSC 05B2005B10
keywords circulantweighingmatricesnonexistenceproofsmultipliertheoremsgroupringsorbittablesproperexhaustivesearchopencasestable
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's goal is to prove that twelve parameter pairs left open in the 2018 update of the standard small-order table for circulant weighing matrices—square matrices with entries $0,\pm1$, constant row weight $k$, and mutually orthogonal cyclic shifts—admit no solution. These cases include $\mathrm{CW}(110,81)$ and $\mathrm{CW}(143,36)$, and closing them reduces the table's unresolved entries from 34 to 22 over the range $n\le200$, $k\le100$. The proofs convert a candidate matrix into a group-ring polynomial, use multiplier theorems to force the nonzero entries to be unions of orbits, and then solve a small system of intersection-number equations that admits no compatible assignment; for the five hardest cases the same orbit framework is settled by an exhaustive backtracking search rather than by hand. The paper also completes the classification of proper weight-$16$ circulant weighing matrices, with orders exactly $21$, $31$, $63$, and $14m$ for every $m\ge2$. If correct, the small-parameter table is sharply narrowed and the known boundary between existent and nonexistent circulant weighing matrices is redrawn.

What carries the argument

The load-bearing machinery is the orbit table for the multiplier group acting on $\mathbb{Z}_n\cong\mathbb{Z}_d\times\mathbb{Z}_m$. Each cell lists the full-group orbits that project onto a given orbit in each factor; assigning an orbit to the positive set $P$, the negative set $N$, or neither fixes row sums $r_i$ and column sums $c_j$, which Lemma 2.5 requires to satisfy $\sum b_i=s$ and $\sum b_i^2=k=s^2$. The equations are small enough to solve by hand or short exhaust, and the table then shows that every alleged solution lifts to something whose group-ring product $A(X)A(X^{-1})$ is not $k$. When the multiplier group is too small for hand work but the search space is still finite, Algorithm 1 performs the same assignment exhaustively by recursing over orbits with the three options 'skip, add to $P$, add to $N$' and pruning partial row and column sums.

What would settle it

Exhibit a $0,\pm1$ circulant matrix with $WW^T=kI_n$ for any of the twelve parameter pairs, for instance a $\mathrm{CW}(144,49)$, and that nonexistence claim falls. For the classification, a proper $\mathrm{CW}(n,16)$ with $n\notin\{21,31,63\}\cup\{14m:m\ge2\}$ would refute Theorem 6.2. A reader can also rerun an independent implementation of Algorithm 1: finding any matrix among the five computer-eliminated cases would overturn them.

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Extended reading notes

Core claim

The central discovery is a set of twelve nonexistence theorems: no circulant weighing matrix exists for $\mathrm{CW}(110,81)$, $\mathrm{CW}(154,81)$, $\mathrm{CW}(130,81)$, $\mathrm{CW}(143,81)$, $\mathrm{CW}(143,36)$, $\mathrm{CW}(132,81)$, $\mathrm{CW}(144,49)$, $\mathrm{CW}(152,49)$, $\mathrm{CW}(160,49)$, $\mathrm{CW}(104,81)$, $\mathrm{CW}(160,81)$, or $\mathrm{CW}(182,64)$. The arguments model a putative matrix as $A(X)$ in the integer group ring of the cyclic group, where the condition is $A(X)A(X^{-1})=k$; multiplier theorems turn the support into a union of orbits, and folding by coprime divisors $d,m$ of $n$ yields equations $\sum b_i=s$ and $\sum b_i^2=k=s^2$ that the orbit structure must satisfy. Several of these cases are eliminated by hand from those equations, while the remaining five are eliminated by the paper's exhaustive orbit search. The classification result is that every proper $\mathrm{CW}(n,16)$—one whose group-ring element is not a multiple of a smaller circulant weighing matrix—has order $21$, $31$, $63$, or $14m$ for some $m\ge2$, and all of those orders occur.

Load-bearing premise

The load-bearing premise is that Algorithm 1's backtracking search is exhaustive and its implementation is free of false negatives, because the five cases $\mathrm{CW}(144,49)$, $\mathrm{CW}(152,49)$, $\mathrm{CW}(160,49)$, $\mathrm{CW}(104,81)$, and $\mathrm{CW}(160,81)$ are settled only by that search, whose code, logs, and certificate outputs are not included in the paper.

Editorial extensions

If this is right

  • The 2018 open-case table for $n\le200$, $k\le100$ goes from 34 unresolved parameter pairs down to 22, with no existence entry disturbed.
  • The infinite family statement in Proposition 3.2 rules out every $\mathrm{CW}(11m,81)$ with $\gcd(33,m)=1$ and $3$ self-conjugate modulo $m$, not just the tabulated cases.
  • Every proper $\mathrm{CW}(n,16)$ now has a known order: $21$, $31$, $63$, or $14m$ for $m\ge2$, so orders outside this set are either improper or nonexistent.
  • For parameters with a sufficiently large multiplier group, Algorithm 1 is a decision procedure: it either produces a circulant weighing matrix or proves none exists, as done for $\mathrm{CW}(144,49)$, $\mathrm{CW}(152,49)$, $\mathrm{CW}(160,49)$, $\mathrm{CW}(104,81)$, and $\mathrm{CW}(160,81)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same orbit-table and search machinery could be exported to the remaining open cases that have a nontrivial multiplier group; the bottleneck there is not search size but the absence of a multiplier theorem strong enough to start the table, so new multiplier results would have the largest effect.
  • The weight-16 classification suggests a template for the next square weights: if the analogous complete classification for weight 25 were verified, the same Type II construction and finiteness results would determine exactly which orders are proper, and the same $14m$ pattern may reappear for other weights.
  • Because the five computer-eliminated cases come with no code, logs, or certificates, an independent reimplementation of Algorithm 1 is a direct way to test the paper's computational claims; the reported 2.4-billion-case run for $\mathrm{CW}(144,49)$ gives a concrete scale for that check.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies circulant weighing matrices CW(n,k) with n ≤ 200 and k ≤ 100, using group-ring methods and multiplier theorems. It claims nonexistence for twelve open cases in Strassler's table: seven by hand using orbit tables and folding arguments, and five more (CW(144,49), CW(152,49), CW(160,49), CW(104,81), CW(160,81), plus CW(182,64) via an ICW search) by a backtracking computer search over orbit assignments. The paper also characterizes the proper CW(n,16), proving that their orders are exactly 21, 31, 63, and 14m for all m ≥ 2.

Significance. If the proofs are correct, the paper resolves twelve previously open cases and completes the classification of proper circulant weighing matrices of weight 16. The orbit-table method is a useful systematic framework, and the hand proofs in Sections 3–4 are largely checkable. The claimed computer-eliminated cases, however, are not independently verifiable from the manuscript alone, which limits the reproducibility of the headline results.

major comments (3)
  1. [Section 5, Algorithm 1 and Table 9] The nonexistence of CW(144,49), CW(152,49), CW(160,49), CW(104,81), CW(160,81), and CW(182,64) rests entirely on the unreleased backtracking implementation described in Section 5. The manuscript provides pseudocode and node counts (e.g., 2.4 billion for CW(144,49)) but no code, input orbit tables, branch order, or logs, and the assertion that "Algorithm 1 will find all CW(n,k)" is not accompanied by a formal completeness proof. A false negative in any of these searches would invalidate six of the twelve headline nonexistence claims, so this is a load-bearing gap. Please make the implementation and search logs available, or provide a machine-checkable certificate and a rigorous completeness argument for the exact search space used.
  2. [Section 5, Algorithm 1 pseudocode] The pruning condition `else if rj = 0` appears to test the row-sum vector r at the moment column j is completed; based on the initialization and on Table 1, this should be `cj`. If the executed code follows the printed pseudocode, the search may prune incorrectly and be incomplete; if it does not, the pseudocode does not describe the computation actually performed. This inconsistency must be corrected before the computational results can be assessed.
  3. [Section 3.5, Proposition 3.5] The proof states that the first two Z13 solutions, (0,3,3,0,-3) and (0,4,1,1,-3), "may be quickly eliminated" because a column sum of -9 is impossible, and then for the third solution it checks only one representative of the 12 choices. Since this proposition is presented as a hand proof, these finite checks should be written out explicitly; as it stands, the elimination is asserted rather than demonstrated, leaving the argument incomplete at a point that is load-bearing for the nonexistence of CW(143,81).
minor comments (4)
  1. [Section 5, Proposition 5.1] The proof line "Since there is no ICW_2(91,81)" contradicts Table 9, which records zero for ICW_2(91,64); the weight 81 should be 64.
  2. [Section 5, opening paragraph] The sentence "The seven cases resolved above leave 27 open cases" is arithmetically inconsistent: Sections 3–4 resolve six cases (CW(110,81), CW(154,81), CW(130,81), CW(143,81), CW(143,36), and CW(132,81)). The count should be 28 open cases, or a seventh resolved case should be identified.
  3. [Section 3.4, Proposition 3.4] The phrase "N has order 36 ≡ 0 (mod 1)2" contains a typographical error; it should read "(mod 12)".
  4. [Table 11 caption] The caption ending "Entries cm are for all m such that cm ≥ k" is garbled; please clarify the notation for infinite families such as 2m, 7 and 14m, 21, 31, 63.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the nonexistence proofs are direct contradictions derived from the defining equation and cited multiplier theorems, with no fitted parameter renamed as a prediction.

full rationale

The derivation chain for the twelve nonexistence results starts from the defining identity AA^(-1)=k, then applies external multiplier theorems (Lander's Theorem 2.2, the classical multiplier theorem Theorem 2.4, McFarland's Theorem 4.1, and the Arasu-Xiang multiplier theorem for CW(143,36)). Each hand proof reduces the orbit structure to the intersection-number equations of Lemma 2.5 and derives an algebraic contradiction; the computer searches in Section 5 enumerate orbit assignments against the exact dot-product condition and are therefore not fitting a parameter from the sought conclusion. The Section 6 classification of proper CW(n,16) relies on the published prior classification [8] for odd orders and the Type II dichotomy; although [8] shares an author with this paper, it is an external published theorem invoked as such, and the new even-order argument is an independent construction and proof rather than a restatement of the conclusion. The main caveats—the lack of shipped code and logs for Algorithm 1 and the apparent ICW_2(91,64) typo in Proposition 5.1—are reproducibility and correctness risks, not circularity. No step renames an empirical pattern, defines an input in terms of the claimed output, or imports a forced choice from an unverified self-citation.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claims rest on cited theorems in group rings and multiplier theory (standard background), and on the completeness of an unshipped computer exhaust for five cases. There are no fitted parameters and no invented entities; the paper is pure existence/nonexistence mathematics.

assumptions (3)
  • standard math Lander/Jungnickel self-conjugacy theorem (Theorem 2.2) and multiplier theorems (Theorems 2.4 and 4.1) apply to the parameter pairs considered.
    Each nonexistence proof uses these cited results to restrict P and N to unions of multiplier orbits; if any application is invalid, the orbit exhaustion is incomplete and the contradiction does not follow. The theorems themselves are established in refs [14], [3], [5], [18].
  • ad hoc to paper Algorithm 1's backtracking search is complete and correctly implemented for the five computer-eliminated cases.
    Section 5 reports computation times and case counts (15 days, 2.4 billion putatives for CW(144,49)) but provides no code or logs. The nonexistence of CW(144,49), CW(152,49), CW(160,49), CW(104,81), and CW(160,81) depends on this unverified exhaust.
  • standard math The Type II characterization from [8] is complete for even-order proper CW(n,16).
    Theorem 6.2 builds directly on [8]'s dichotomy (odd orders 21,31,63 or Type II); the new proof only rules out Type II sources beyond CW(7,4).

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Pith. "Pith review of New Nonexistence Results on Circulant Weighing Matrices." pith.science (2026). https://pith.science/paper/WG2QBMDZ

@misc{pith2026190808447,
  author       = {Pith},
  title        = {Pith review of: New Nonexistence Results on Circulant Weighing Matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WG2QBMDZ}},
  note         = {Machine review of arXiv:1908.08447}
}
abstract

A circulant weighing matrix $W = (w_{i,j})$ is a square matrix of order $n$ and entries $w_{i,j}$ in $\{0, \pm 1\}$ such that $WW^T=kI_n$. In his thesis, Strassler gave a table of existence results for such matrices with $n \leq 200$ and $k \leq 100$. In the latest version of Strassler's table given by Tan \cite{arXiv:1610.01914} there are 34 open cases remaining. In this paper we give nonexistence proofs for 12 of these cases, report on preliminary searches outside Strassler's table, and characterize the known proper circulant weighing matrices.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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