[#P2878] Orders of ternary row-orthogonal matrices with a full row
Problem. Let \(n\ge 1\). Suppose there is an \(n\times n\) matrix \(A\) with entries in \(\{-1,0,1\}\) whose rows are nonzero and pairwise orthogonal over \(\mathbb R\), and one row of \(A\) is the all-ones vector. Must \(n\) equal \(1\), \(2\), or a multiple of \(4\)?
1Context
The normalized row supports convert the problem into reciprocal-sum and signed-intersection constraints. Certified exclusions by order and feasible support multisets are reusable for both computational and arithmetic approaches.
2Problem setup
Definition 1 (Rows are pairwise orthogonal when their standard Euclidean dot product). Rows are pairwise orthogonal when their standard Euclidean dot product is zero for every two distinct rows.
Definition 2 (A full row has no zero entries; multiplying its columns by signs lets any full row be normalized to the all-ones vector without changing row orthogonality). A full row has no zero entries; multiplying its columns by signs lets any full row be normalized to the all-ones vector without changing row orthogonality.
Remark 1. The normalized row supports convert the problem into reciprocal-sum and signed-intersection constraints. Certified exclusions by order and feasible support multisets are reusable for both computational and arithmetic approaches.
3What counts as a solution
- Prove that no such matrix exists when n is odd or congruent to 2 modulo 4, or give an explicit counterexample of one of those orders.
- A computational counterexample must include the full matrix. A finite nonexistence extension must include a proof-producing SAT, ILP, or exhaustive-search certificate with row and sign symmetries specified.
1Status
Current status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources. Prove that no such matrix exists when n is odd or congruent to 2 modulo 4, or give an explicit counterexample of one of those orders.[1]
1Records
Notes and companion material
Original intake status. UNKNOWN as of 2026-07-31. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources.
- On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for question 446995; the full comment chain was reviewed because several proposed proofs were later corrected.
- Christian and Shader, Electronic Journal of Combinatorics 11 (2004), N1, study exactly row-orthogonal (0,1,-1)-matrices with a full row and leave the 1,2,0 mod 4 order classification as a question.
- The comments report independent ILP and clique-search exclusions through n=20. They also record denominator arguments for odd prime powers and twice odd prime powers, consistent with the 2004 paper.
- A TheoremDB search for ternary row-orthogonal matrices, full rows, and the 1,2,4k order condition found no duplicate.
Recorded example 1. At n=4, the rows (1,1,1,1), (1,1,-1,-1), (1,-1,0,0), and (0,0,1,-1) form an admissible matrix.
Computational notes
- The MathOverflow comments report exact nonexistence checks through n=20. The first unchecked composite orders outside the families handled in the 2004 paper should be independently audited before extending that range.
How the 2 records connect
ProblemOrders of ternary row-orthogonal matrices with a full row
2See also
- Components of the Pasch-switch graph on STS(15) classesdesign theory
- Hadamard matrix conjecturedesign theory
- Largest cyclic 3-(31,5,1) packingdesign theory
How to cite
TheoremDB contributors, “Orders of ternary row-orthogonal matrices with a full row,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/ternary-orthogonal-full-row-orderThis page as plain text: ternary-orthogonal-full-row-order.md
This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net[1], current as of July 31, 2026.
1References
- Packet source. MathOverflow: Orthogonal vectors with entries from {-1,0,1}. Question 446995 and all visible comments, checked through the Stack Exchange API on 2026-07-27. Question 446995 and all visible comments, checked through the Stack Exchange API on 2026-07-27. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Orders of ternary row-orthogonal matrices with a full row: UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources.Source named by the research packet.
- Justin D. Christian and Bryan L. Shader, “Nonexistence Results for Hadamard-Like Matrices,” Electronic Journal of Combinatorics 11 (2004), Note N1. Status evidence identified in the source record and checked at the linked publication. ↗website · primary source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources.Also cited at abstract and Theorems 7 through 9 on forbidden orders p^k, 2p^k, and 3p.Source used to assess the problem's recorded status.For Orders of ternary row-orthogonal matrices with a full row, this source proves the cited arithmetic nonexistence families for row-orthogonal ternary matrices with a full row.
- Arun, “Reference Request: Maximal Determinant of Matrices with Pairwise Orthogonal Rows and Entries in {1, 0, -1},” MathOverflow question 418009, asked March 12, 2022, checked 2026-08-01. Status evidence identified in the source record and checked at the linked publication. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources.Also cited at question 418009, checked through the Stack Exchange API on 2026-08-01.Source used to assess the problem's recorded status.For Orders of ternary row-orthogonal matrices with a full row, this source asks a related maximal-determinant reference question and does not settle which orders admit a full-row ternary orthogonal matrix.
This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.