[#P2640] Maximum determinant of a skew Seidel matrix of order 34

Problem. Determine \(\max\det S\), where \(S\) ranges over the \(34\times34\) skew-symmetric matrices with zero diagonal and off-diagonal entries in \(\{-1,1\}\).

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Definitions and notation

1Problem setup

Definition 1. Skew-symmetric means S^T=-S.

Remark 1. Such matrices encode tournaments; switching and relabeling preserve the determinant.

2What counts as a solution

Provide a skew Seidel matrix attaining the maximum and an exact upper-bound certificate over all switching classes.

1Status

Saved packet · July 25, 2026

What counts as a solution

Saved packet status (The maximum is between 35^16 and 6,876,227,375,063^2). A principal submatrix of a skew conference matrix gives the lower endpoint, while the skew Ehlich-Wojtas bound and Pfaffian integrality give the upper endpoint.^[1]

1Packet records

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Record	Kind	Assessment
A principal submatrix of a skew conference matrix gives the lower endpoint, while the skew Ehlich-Wojtas bound and Pfaffian integrality give the upper endpoint.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1 A principal submatrix of a skew conference matrix gives the lower endpoint, while the skew Ehlich-Wojtas bound and Pfaffian integrality give the upper endpoint.^[1] Relevance to this problem For Maximum determinant of a skew Seidel matrix of order 34, record ssm34-claim-certified-interval (“The maximum is between 35^16 and 6,876,227,375,063^2”) records a bound, answer, status fact, or structural consequence. The record states: A principal submatrix of a skew conference matrix gives the lower endpoint, while the skew Ehlich-Wojtas bound and Pfaffian integrality give the upper endpoint. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state supported Scope all 34 by 34 skew-symmetric matrices with zero diagonal and every off-diagonal entry in {-1,1} Sources Reference 1 Details Write \(D_{34}\) for the requested maximum. The certified bounds are \[ 5{,}070{,}942{,}774{,}902{,}496{,}337{,}890{,}625 =35^{16} \leq D_{34} \leq 6{,}876{,}227{,}375{,}063^2 =47{,}282{,}502{,}913{,}565{,}795{,}274{,}253{,}969. \] The lower endpoint is attained by the explicit order-34 matrix in `ssm34-artifact-gs36-principal-minor`. The upper endpoint follows for every matrix in the stated family. These sources and the computation do not determine equality at either endpoint, so the exact maximum remains open in this entry. Open full record →
Deleting two matching rows and columns from the Goethals-Seidel skew conference matrix of order 36 gives Pfaffian -35^8.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 2 Deleting two matching rows and columns from the Goethals-Seidel skew conference matrix of order 36 gives Pfaffian -35^8.^[2] Relevance to this problem For Maximum determinant of a skew Seidel matrix of order 34, record ssm34-claim-goethals-seidel-lower-bound (“An explicit order-34 matrix has determinant 35^16”) records a bound, answer, status fact, or structural consequence. The record states: Deleting two matching rows and columns from the Goethals-Seidel skew conference matrix of order 36 gives Pfaffian -35^8. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state established Scope the order-34 principal submatrix obtained by deleting indices 0 and 1 from the Goethals-Seidel order-36 skew conference matrix Sources Reference 2 Details Goethals and Seidel give four circulant first rows of length nine: \[ \begin{aligned} a&=(0,+,+,-,+,-,+,-,-),\\ b&=(+,-,+,+,-,-,+,+,-),\\ c&=(-,-,+,+,+,+,+,+,-),\\ d&=(+,+,+,-,+,+,-,+,+). \end{aligned} \] Let \(A,B,C,D\) be the corresponding circulants, let \(R\) be the reversal matrix, and use their four-by-four block array. The replay verifies that the resulting \(H\) has entries in \(\{-1,1\}\), satisfies \(H+H^T=2I\), and obeys \(HH^T=36I\). Thus \(K=H-I\) is a skew conference matrix with \(KK^T=35I\). Delete rows and columns 0 and 1 from \(K\). The remaining matrix \(S\) is a skew Seidel matrix of order 34. Exact Pfaffian elimination gives \[ \operatorname{pf}(S)=-2{,}251{,}875{,}390{,}625=-35^8, \] and fraction-free Bareiss elimination gives \[ \det S=5{,}070{,}942{,}774{,}902{,}496{,}337{,}890{,}625=35^{16}. \] This is also the order-34 case of the principal-submatrix lower bound in Theorem 5.5 of Klanderman et al. Related: Computation 1 Open full record →
The general order 2 modulo 4 bound gives 65 times 31^16, and the odd-square constraint rounds it down.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 The general order 2 modulo 4 bound gives 65 times 31^16, and the odd-square constraint rounds it down.^[1] Relevance to this problem For Maximum determinant of a skew Seidel matrix of order 34, record ssm34-claim-pfaffian-rounded-upper-bound (“Every determinant is at most 6,876,227,375,063^2”) records a bound, answer, status fact, or structural consequence. The record states: The general order 2 modulo 4 bound gives 65 times 31^16, and the odd-square constraint rounds it down. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope every 34 by 34 skew Seidel matrix Sources Reference 1 Argument Klanderman et al. define \(M(n)=\max\sqrt{\det S}\). Their Theorem 5.4, which collects the skew Ehlich-Wojtas result, gives \[ M(n)\leq\sqrt{2n-3}\,(n-3)^{(n-2)/4} \] when \(n\equiv2\pmod4\). At \(n=34\), this says \[ \det S\leq65\,31^{16} =47{,}282{,}502{,}913{,}567{,}042{,}148{,}851{,}265. \] Cayley's Pfaffian identity says \(\det S=\operatorname{pf}(S)^2\), and parity makes the Pfaffian odd. Exact integer square root gives \[ \left\lfloor\sqrt{65\,31^{16}}\right\rfloor =6{,}876{,}227{,}375{,}063. \] This integer is odd, while the next odd integer already has square larger than \(65\,31^{16}\). Hence every admissible determinant is at most \[ 6{,}876{,}227{,}375{,}063^2 =47{,}282{,}502{,}913{,}565{,}795{,}274{,}253{,}969. \] The equality form in the unrounded bound would require \(2n-3\) to be a square. Here \(65\) is not a square, which independently rules out equality in that real-valued bound. Related: Computation 1 Open full record →
Standard-library Python reconstructs the published circulant blocks and checks every claimed identity with exact arithmetic.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Standard-library Python reconstructs the published circulant blocks and checks every claimed identity with exact arithmetic.^[2] Relevance to this problem For Maximum determinant of a skew Seidel matrix of order 34, record ssm34-artifact-gs36-principal-minor (“Exact generator, Pfaffian, and determinant verifier”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python reconstructs the published circulant blocks and checks every claimed identity with exact arithmetic. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope the Goethals-Seidel order-36 block construction and its principal submatrix on indices 2 through 35 Sources Reference 2 Entry point `Join source_lines with LF characters and execute the resulting Python program` Runtime Python 3.10 or newer, standard library Details The program fixes the circulant convention by setting entry \((i,j)\) to first-row entry \((j-i)\bmod9\). It builds the Goethals-Seidel array, checks all 1,296 entries of \(H+H^T=2I\) and \(HH^T=36I\), then checks the skew Seidel conditions after deletion. The compact matrix certificate is the SHA-256 digest of its upper-triangle signs in row-major order, writing `+` for 1 and `-` for -1. The order-34 digest is `c081b626a079cca244e06357bacacad2a406f7512557c554a5dc78e0477bcd35`. Exact rational Pfaffian elimination and integer Bareiss elimination independently agree through \(\det S=\operatorname{pf}(S)^2\). The stable output digest is recorded with the program. Related: Computation 2 Open full record →
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Notes and companion material

Local search generates incumbents quickly, while proving optimality requires partitioned computation whose failed regions can be shared.

Original intake status. A 2024 primary paper studies this maximum for general even order and supplies upper bounds. Its abstract does not state the exact order-34 value; a full table and citation check remains necessary.

Canonicalize tournaments under vertex relabeling and switching. Determinants are squares of integer Pfaffians, which gives a strong exact consistency check.
Trap: the Hadamard bound 33^17 ignores skew-symmetry and is far too loose. Floating-point log determinants are useful only for ranking.

Recorded example 1. The best sampled matrix is encoded by the 561 upper-triangle signs in row-major order under seed 1534.

Computational notes

Among 20000 seeded random tournaments, the best floating-point score was replayed with exact Bareiss elimination. Its determinant was 1445221973536022889601=38016075199^2, confirming the required Pfaffian-square check.

How the 4 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemMaximum determinant of a skew Seidel matrix of order 34

All 3 recorded relations between these records and the problem

2See also

Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20matrix theory
A Hadamard matrix of order 668matrix theory
Hadamard matrix conjecturematrix theory

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Plain text

“Maximum determinant of a skew Seidel matrix of order 34.” TheoremDB. P2640. Problem statement; statement text SHA-256 247e92f86dee957af3f87dde04887ae7058954ac4fc64c5b2250bc9a954c323d. https://theoremdb.org/statement/?ref=P2640

BibTeX

@misc{theoremdb-problem-247e92f86dee957af3f87dde04887ae7058954ac4fc64c5b2250bc9a954c323d,
  title = {{Maximum determinant of a skew Seidel matrix of order 34}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 247e92f86dee957af3f87dde04887ae7058954ac4fc64c5b2250bc9a954c323d},
  url = {https://theoremdb.org/statement/?ref=P2640}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from arxiv.org^[1], current as of July 25, 2026.

1Lean verification

1References

Packet source. Sarah Klanderman, MurphyKate Montee, Andrzej Piotrowski, Alex Rice, and Bryan Shader, Determinants of Seidel Tournament Matrices, arXiv:2406.09697v1 (2024). Klanderman, Montee, Piotrowski, Rice, and Shader, Theorems 5.4 and 5.5, combined with the order-36 construction of Goethals and Seidel and the exact replay below. ↗preprint · primary source · arXiv:2406.09697v1 · checked 2026-08-01Source use: original summary.Provides the skew Ehlich-Wojtas determinant bound and Pfaffian constraints used for the upper endpoint.Also cited at Klanderman et al., Theorem 5.4(a,d); Greaves and Suda, arXiv:1601.02769, Theorem 1.1 and the skew EW matrix discussion; Cayley's Pfaffian identity.Source named by the research packet.For Maximum determinant of a skew Seidel matrix of order 34: The general order 2 modulo 4 bound gives 65 times 31^16, and the odd-square constraint rounds it down.
J. M. Goethals and J. J. Seidel, A skew Hadamard matrix of order 36, Journal of the Australian Mathematical Society 11(3) (1970), 343-344. Goethals and Seidel, A skew Hadamard matrix of order 36, Theorems 1 and 2, pages 343-344; exact replay in ssm34-artifact-gs36-principal-minor. ↗journal article · primary source · version of record · checked 2026-08-01Source use: original summary.Gives the order-36 skew conference matrix whose principal submatrix attains determinant 35^16.Also cited at Inline Python 3 standard-library replay of Goethals and Seidel, executed on 2026-07-25.For Maximum determinant of a skew Seidel matrix of order 34: Standard-library Python reconstructs the published circulant blocks and checks every claimed identity with exact arithmetic.

CC0 tournament determinant target with a primary-source check and exact random-search incumbent.

[#P2640] Maximum determinant of a skew Seidel matrix of order 34

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1Problem setup

2What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance