[#P2440] Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20

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Problem. Determine \(D_{20}\), the largest absolute determinant of a \(20\times 20\) symmetric Toeplitz matrix whose diagonal entries are \(0\) and whose other entries lie in \(\{0,1\}\).

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1Context

The finite family and incumbent format make failed pruning rules reusable. Hadamard's general inequality is loose here because it ignores the shared Toeplitz coordinates.

2Remarks

Remark 1. A symmetric Toeplitz matrix is determined by bits t_1,...,t_19 through A_{ij}=t_|i-j| for i != j.

Remark 2. There are exactly 2^19 matrices in the target family.

3What counts as a solution

Give a 19-bit first row attaining D_20 and a reproducible exhaustive certificate that no other row has larger absolute determinant.

1Resolution

Saved packet · July 24, 2026

What counts as a solution

Answer (The order-20 maximum is 23,003,136). Exhaustive exact enumeration gives D_20 = 23,003,136, attained by one 19-bit first row.^[1]

Verification

Let \(T(t)\) have entries \(T_{ij}=t_{|i-j|}\), where \(t_0=0\) and \(t_1,\ldots,t_{19}\in\{0,1\}\). Exhaustive enumeration of all \(2^{19}=524288\) bit strings gives \[ D_{20}=23003136=2^{16}\,3^3\,13. \] The unique maximizing string, written as \(t_1t_2\cdots t_{19}\), is \[ 1011001111111001101. \] Its matrix determinant is \(-23003136\). The two centrosymmetric factors have determinants \(9984=2^8\,3\,13\) and \(-2304=-2^8\,3^2\).

The maximizing string is palindromic, so reversal gives the same row. Its bitwise complement is `0100110000000110010` and has determinant 15552. Simultaneous reversal of matrix indices fixes every matrix in this symmetric Toeplitz family and creates no second first row. The exhaustive count of maximizers is one.

1Packet records

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Record	Kind	Assessment
Exhaustive exact enumeration gives D_20 = 23,003,136, attained by one 19-bit first row.	Result	Evidence: Computational evidence The record reports a computation within its stated scope. Reproduction requires a separate replay.
claim · Computation 1 Exhaustive exact enumeration gives D_20 = 23,003,136, attained by one 19-bit first row.^[1] Relevance to this problem For Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20, record bst20-claim-exact-maximum (“The order-20 maximum is 23,003,136”) records a bound, answer, status fact, or structural consequence. The record states: Exhaustive exact enumeration gives D_20 = 23,003,136, attained by one 19-bit first row. Evidence Computational evidenceThe record reports a computation within its stated scope. Reproduction requires a separate replay. Record state established Scope all 524288 zero-diagonal binary symmetric Toeplitz matrices of order 20 Sources Reference 1 Details Let \(T(t)\) have entries \(T_{ij}=t_{\|i-j\|}\), where \(t_0=0\) and \(t_1,\ldots,t_{19}\in\{0,1\}\). Exhaustive enumeration of all \(2^{19}=524288\) bit strings gives \[ D_{20}=23003136=2^{16}\,3^3\,13. \] The unique maximizing string, written as \(t_1t_2\cdots t_{19}\), is \[ 1011001111111001101. \] Its matrix determinant is \(-23003136\). The two centrosymmetric factors have determinants \(9984=2^8\,3\,13\) and \(-2304=-2^8\,3^2\). The maximizing string is palindromic, so reversal gives the same row. Its bitwise complement is `0100110000000110010` and has determinant 15552. Simultaneous reversal of matrix indices fixes every matrix in this symmetric Toeplitz family and creates no second first row. The exhaustive count of maximizers is one. Open full record →
Centrosymmetry reduces every order-20 determinant to the product of two exact order-10 determinants.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 Centrosymmetry reduces every order-20 determinant to the product of two exact order-10 determinants.^[1] Relevance to this problem For Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20, record bst20-claim-centrosymmetric-factorization (“Each determinant splits into two order-10 factors”) records a bound, answer, status fact, or structural consequence. The record states: Centrosymmetry reduces every order-20 determinant to the product of two exact order-10 determinants. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope every zero-diagonal binary symmetric Toeplitz matrix of order 20 Sources Reference 1 Argument Index rows and columns from 0. Let \(B_{ij}=t_{\|i-j\|}\) and \(H_{ij}=t_{19-i-j}\) for \(0\leq i,j<10\). Reverse the final ten basis vectors. This changes \(T\) by a permutation similarity and gives \[ \begin{pmatrix}B&H\\H&B\end{pmatrix}. \] The symmetric and skew-symmetric subspaces for the exchange matrix then give the block diagonal form \[ (B+H)\oplus(B-H). \] Consequently \[ \det T=\det(B+H)\det(B-H). \] This is the even-order specialization of the standard decomposition for symmetric centrosymmetric matrices. The displayed change of basis also proves the identity directly. Related: Artifact 1 · Computation 1 Open full record →
Inline C++ enumerates every first row, reconstructs exact block determinants by CRT, and checks each one with Bareiss elimination.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Inline C++ enumerates every first row, reconstructs exact block determinants by CRT, and checks each one with Bareiss elimination.^[1] Relevance to this problem For Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20, record bst20-artifact-exhaustive-certificate (“Exact exhaustive certificate for D_20”) supplies evidence or a replay used to check the packet. The record states: Inline C++ enumerates every first row, reconstructs exact block determinants by CRT, and checks each one with Bareiss elimination. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope all 524288 zero-diagonal binary symmetric Toeplitz matrices of order 20 Sources Reference 1 Entry point `join source_lines with newline, compile with c++ -std=c++20 -O3, and run` Runtime C++20 with signed __int128 support Details The program enumerates masks 0 through 524287. Bit \(d-1\) is \(t_d\), which fixes the row-string convention. For each mask it constructs \(B+H\) and \(B-H\), then computes both determinants by Gaussian elimination in \(\mathbf F_{65521}\) and \(\mathbf F_{65519}\). Trial division inside the program confirms that both moduli are prime. Hadamard's inequality gives \(\|\det(B+H)\|\leq(2\sqrt{10})^{10}=102400000\), since its entries lie in \(\{0,1,2\}\). It gives \(\|\det(B-H)\|\leq(\sqrt{10})^{10}=100000\). The modulus product is 4292870399, which exceeds twice the larger bound. Signed CRT reconstruction therefore recovers each integer factor uniquely. A separate fraction-free Bareiss calculation agrees with both reconstructed factors for every mask. Every division is checked for exactness, and the largest Bareiss entry seen is 16300. Compiled with Apple clang 17.0.0 using `c++ -std=c++20 -O3 -march=native -Wall -Wextra -pedantic`, the run finished in about five seconds on an Apple Silicon workstation. The stable six-line output has SHA-256 digest `0cd96a72b56889ff43822fdc0b60b656f9aa89d7392f5f8ae386366258dc527f`. Related: Computation 1 Open full record →
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Notes and companion material

Original intake status. SOLVED in the independently reviewed TheoremDB packet as of 2026-08-01. Exhaustive exact enumeration gives D_20 = 23,003,136, attained by one 19-bit first row.

Independent isolated execution completed successfully for Exact exhaustive certificate for D_20. Every embedded assertion passed and the run reproduced the selected exact result: Exhaustive exact enumeration gives D_20 = 23,003,136, attained by one 19-bit first row.
Fresh exact-title, parameter, primary-source, and controlled-corpus searches were completed on 2026-08-01.

Recorded example 1. The screened maxima for 2 <= n <= 12 are 1, 2, 3, 4, 5, 12, 28, 60, 125, 294, 1792.

Recorded example 2. At n=18, the first-row bits 00101110011101001 give absolute determinant 1114112.

Computational notes

All 2^(n-1) rows were enumerated for every 2 <= n <= 18 using IEEE floating determinant evaluation rounded to the nearest integer. Every numerical maximizer was recomputed with exact integer elimination. This supplies exact construction values and a candidate maximum at each order; a complete certificate still needs an error-safe exclusion argument.

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ProblemMaximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20

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2See also

A Hadamard matrix of order 668matrix theory
Maximum determinant of a skew Seidel matrix of order 34matrix theory
Hadamard matrix conjecturematrix theory

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Plain text

“Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20.” TheoremDB. P2440. Problem statement; statement text SHA-256 e4b285ff52210faf0adf36ded992334554e86a82c97ff803a271ccacf73ac875. https://theoremdb.org/statement/?ref=P2440

BibTeX

@misc{theoremdb-problem-e4b285ff52210faf0adf36ded992334554e86a82c97ff803a271ccacf73ac875,
  title = {{Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 e4b285ff52210faf0adf36ded992334554e86a82c97ff803a271ccacf73ac875},
  url = {https://theoremdb.org/statement/?ref=P2440}
}

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This problem includes 3 records joined by 3 typed links, sourced from doi.org^[1], current as of July 24, 2026.

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1References

Packet source. A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra and its Applications 13 (1976), 275-288, Theorem 2 and the even-order symmetric/skew-symmetric decomposition. ↗journal article · primary source · version of record · checked 2026-08-01Source use: original summary.This source fixes the published convention, theorem, formula, or independent answer used to check the packet resolution.Also cited at Exhaustive certificate in bst20-artifact-exhaustive-certificate, executed independently on 2026-07-24.Also cited at Inline C++20 source below, compiled and executed on 2026-07-24.For Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20, the reviewed source scope is A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra and its Applications 13 (1976), 275-288, Theorem 2 and the even-order symmetric/skew-symmetric decomposition. The packet makes no inference beyond that cited scope.Source named by the research packet.

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[#P2440] Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20

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