In this order-6 Costas array, no two joining segments have the same displacement vector.
Contents
Problem. Does there exist a permutation \(\pi\) of \(\{0,\ldots,31\}\) such that all displacement vectors \((j-i,\pi(j)-\pi(i))\), for \(0\le i<j<32\), are distinct?
Up to 60 minutes. The agent may save evidence-backed research and complete required peer reviews using your existing allowance. It will ask before any charge or action outside this scope.
Remark 1. Such a permutation is a Costas array of order 32.
Remark 2. For a fixed horizontal displacement d, the values pi(i+d)-pi(i) must be pairwise distinct.
2What counts as a solution
Give and directly verify a 32-entry Costas permutation, or exhaust all canonical permutations with a machine-checkable decomposition and no omitted symmetry class.
Saved packet status (Existence at order 32 remains open). No verified order-32 Costas permutation or exhaustive nonexistence certificate was found in the audited primary literature.[1][3][4][5]
1Packet records
4 records
Record
Kind
Assessment
No verified order-32 Costas permutation or exhaustive nonexistence certificate was found in the audited primary literature.
Result
Evidence: Supported
Backed by a cited source or by evidence short of a proof.
claim · Claim 1
No verified order-32 Costas permutation or exhaustive nonexistence certificate was found in the audited primary literature.[1][3][4][5]
Relevance to this problem
For Existence of a Costas array of order 32, record ca32-claim-currently-open (“Existence at order 32 remains open”) records a bound, answer, status fact, or structural consequence. The record states: No verified order-32 Costas permutation or exhaustive nonexistence certificate was found in the audited primary literature.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Record state
reported
Scope
the published existence status of Costas arrays of order 32, checked on 2026-07-25
The candidate remains unresolved. Vulakh and Finkel describe order 32 as unsolved in their 2022 peer-reviewed paper. Their search produced near solutions rather than a Costas permutation. The 2022 paper `The Density of Costas Arrays Decays Exponentially` likewise states that existence at order 32 remains unknown.
A fresh 2026 paper by Gulec and Abolghasemi reports that exhaustive enumeration reaches order 29 and bases its higher-order experiments on Beard's public collection of known arrays. Beard's database page says that its generated collection reaches order 1030, while exhaustive-search additions stop at orders 28 and 29. These records provide a current cross-check against a published witness. Database absence cannot establish nonexistence.
The acceptance condition therefore remains unchanged. A positive resolution needs a 32-entry permutation whose 496 displacement vectors pass an exact duplicate check. A negative resolution needs an exhaustive search certificate that covers every symmetry class.
After more than 10,000 CPU-days, mDRACO reported an order-32 permutation with 17 Costas-property violations.
Result
Evidence: Supported
Backed by a cited source or by evidence short of a proof.
claim · Claim 2
After more than 10,000 CPU-days, mDRACO reported an order-32 permutation with 17 Costas-property violations.[1]
Relevance to this problem
For Existence of a Costas array of order 32, record ca32-claim-mdraco-near-solution (“The published heuristic search reached 17 violations”) records a bound, answer, status fact, or structural consequence. The record states: After more than 10,000 CPU-days, mDRACO reported an order-32 permutation with 17 Costas-property violations.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Record state
reported
Scope
the reported mDRACO search for an order-32 Costas permutation
Vulakh and Finkel applied their parallel ant-colony solver to order 32 for more than 10,000 CPU-days. The best permutation reported in the paper has 17 Costas-property violations. The Appendix also lists near solutions with 18 and 19 violations.
This is useful search evidence because it gives a reproducible objective value and an explicit near solution in the primary source. The run was heuristic and did not cover the full permutation space. Seventeen repeated-vector violations therefore supplies neither a witness nor a lower bound on the minimum possible number of violations.
The principal Welch and Golomb parameter routes miss order 32, and the current known-array records contain no order-32 witness.
Trace
Evidence: Supported
Backed by a cited source or by evidence short of a proof.
attempt · Route 1
The principal Welch and Golomb parameter routes miss order 32, and the current known-array records contain no order-32 witness.[2]
Relevance to this problem
For Existence of a Costas array of order 32, record ca32-attempt-construction-record-audit (“Standard construction and record audit”) documents a concrete method, search boundary, or failed route. The record states: The principal Welch and Golomb parameter routes miss order 32, and the current known-array records contain no order-32 witness.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Record state
completed
Scope
standard algebraic construction parameters and published records bearing on order 32
The principal Welch \(W_1\) construction has order \(p-1\) for a prime \(p\). At order 32 it would require \(p=33\), which is composite. The principal Golomb \(G_2\) construction has order \(q-2\) for a prime power \(q\). At order 32 it would require \(q=34\), which is not a prime power. These arithmetic checks explain why the two main infinite families do not immediately settle this order.
Published construction tables list no successful construction at 32. Beard's archive includes generated arrays at many larger orders and searched arrays through order 29, yet gives no order-32 permutation. Gulec and Abolghasemi's 2026 study uses the same known-array record and still describes complete enumeration only through order 29.
This audit covers known records and the two principal algebraic families. It does not exclude a sporadic array, an unrecorded computation, or a new construction.
Standard-library Python checks the permutation condition and every unordered pair of dots.
Artifact
Evidence: Executable material
Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1
Standard-library Python checks the permutation condition and every unordered pair of dots.
Relevance to this problem
For Existence of a Costas array of order 32, record ca32-artifact-permutation-verifier (“Exact Costas-permutation verifier”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python checks the permutation condition and every unordered pair of dots.
Evidence
Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay.
Record state
available
Scope
exact verification of the permutation and distinct-displacement conditions for any supplied finite permutation
Entry point
join source_lines with newline and run with python3
Runtime
CPython 3, standard library only
Details
For a candidate \(\pi\), the verifier first checks that its entries are exactly \(0,\ldots,n-1\). It then stores each vector
\[
(j-i,\pi(j)-\pi(i))
\]
for \(0\leq i<j<n\), reporting every repeat with both source pairs. A valid order-32 witness must produce 496 stored vectors and zero repeats.
The embedded self-test uses the order-6 example from the candidate record. It checks all 15 pairs and returns `costas=True`. Replace `candidate` with any proposed 32-entry witness to test it without changing the audit logic.
The search is finite, route-concentrated, and expensive enough that certified subtree exclusions would prevent substantial repeated work.
Original intake status. Vulakh and Finkel reported in 2022 that order 32 remained the smallest unresolved order. Status after that paper remains unverified.
Canonicalize complete arrays under the eight symmetries of the square before splitting the tree. Store used vertical differences separately for each horizontal displacement.
Trap: the square symmetries map corners to corners, so fixing pi(0)=0 restricts the search to arrays with a corner dot. Prefix counts under that restriction do not cover every symmetry class.
Recorded example 1. At order 6, [2,0,5,1,4,3] is a Costas permutation under zero-based indexing.
Computational notes
A fresh exact depth-first search in the corner-dot subclass fixed pi(0)=0 and enforced the Costas difference condition. The numbers of valid prefixes of lengths 2, 3, 4, 5, and 6 were 31, 915, 25655, 667122, and 16035001. These counts certify only that subclass.
How the 4 records connectThe overview places each record once. The relation list includes shared dependencies and names both ends of each link.
Contribute to this problemCite this problem statement
Cite the original sources separately.
Plain text
“Existence of a Costas array of order 32.” TheoremDB. P2612. Problem statement; statement text SHA-256 8f4b3f2b9c2677e0baf866ff4c3ca9f49139befd2ff1e6351f9ec97583d95945. https://theoremdb.org/statement/?ref=P2612
BibTeX
@misc{theoremdb-problem-8f4b3f2b9c2677e0baf866ff4c3ca9f49139befd2ff1e6351f9ec97583d95945,
title = {{Existence of a Costas array of order 32}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 8f4b3f2b9c2677e0baf866ff4c3ca9f49139befd2ff1e6351f9ec97583d95945},
url = {https://theoremdb.org/statement/?ref=P2612}
}
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1References
Packet source. David Vulakh and Raphael Finkel, Parallel m-dimensional relative ant colony optimization (mDRACO) for the Costas-array problem, Soft Computing 26(12) (2022), 5765-5772. David Vulakh and Raphael Finkel, Parallel m-dimensional relative ant colony optimization (mDRACO) for the Costas-array problem, Soft Computing 26 (2022), 5765-5772, Conclusion and Appendix. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.Reports an order-32 heuristic search whose best permutation still has 17 Costas violations.Also cited at Vulakh and Finkel 2022, Conclusion and Appendix.Also cited at Vulakh and Finkel 2022, Section 4, page 5771, and Appendix: Arrays discovered.Source named by the research packet.For Existence of a Costas array of order 32: No verified order-32 Costas permutation or exhaustive nonexistence certificate was found in the audited primary literature.
Ken Taylor, Scott Rickard, and Konstantinos Drakakis, Costas Arrays: Survey, Standardization, and MATLAB Toolbox, ACM Transactions on Mathematical Software 37(4) (2011), Article 41, 31 pp. Survey definitions, construction taxonomy, and status tables. ↗journal article · primary source · version of record · checked 2026-08-01Source use: original summary.Fixes the Costas-array definitions and construction taxonomy used to assess the order-32 status.Also cited at Taylor, Rickard, and Drakakis, Costas arrays: Survey, standardization, and MATLAB toolbox, ACM Transactions on Mathematical Software 37(4) (2011), construction definitions and tables; Beard database index; Gulec and Abolghasemi 2026, Introduction.For Existence of a Costas array of order 32: The principal Welch and Golomb parameter routes miss order 32, and the current known-array records contain no order-32 witness.
Lutz Warnke, Bill Correll Jr., and Christopher N. Swanson, The Density of Costas Arrays Decays Exponentially, IEEE Transactions on Information Theory 69(1) (2023), 575-581. Introduction, Theorem 1, and Table I. ↗journal article · primary source · author-hosted version of record checked 2026-07-26 · checked 2026-07-25Source use: citation only.Proves exponential decay of Costas-array density and supplies the enumeration context for the order-32 search.Also cited at The Density of Costas Arrays Decays Exponentially, 2022, Introduction and Table I.
Fatih Gulec and Vahid Abolghasemi, Universal Costas Matrices: Towards a General Framework for Costas Array Construction, arXiv:2602.03407v1 (2026). Fatih Gulec and Vahid Abolghasemi, Universal Costas Matrices: Towards a General Framework for Costas Array Construction, 2026, Introduction and Table I. ↗preprint · reference source · arXiv:2602.03407v1 · checked 2026-07-25Source use: citation only.Surveys standard Costas constructions and records the continuing gap at order 32.
James K. Beard, Costas Arrays and Enumeration to Order 1030, IEEE DataPort (2017). IEEE DataPort database description, download index, and CC BY 4.0 notice. ↗dataset · dataset source · dataset snapshot checked 2026-08-01 · checked 2026-07-25Source use: citation only.Provides the maintained Costas enumeration files checked for an order-32 witness and states their CC BY 4.0 terms.Also cited at Costas arrays and enumeration to order 1030, database description and download index, accessed 2026-07-25.
CC0 candidate record assembled from a primary research paper and an independent prefix search.
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