Costas arrays of order 32 and 33

Find a Costas array of order 32 or 33, the smallest orders for which none is known, or extend the complete enumeration of Costas arrays beyond order 29.

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@misc{cairn-costas-arrays-order-32,
  title        = {Costas arrays of order 32 and 33},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/costas-arrays-order-32}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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The problem

A Costas array of order n is an n×n permutation matrix in which the n(n−1)/2 displacement vectors between pairs of 1s are all distinct. Equivalently, a permutation π of {1..n} such that for every shift h, the differences π(i+h) − π(i) are pairwise distinct. They are used in radar and sonar and are closely related to Golomb rulers.

Known status. Algebraic constructions give arrays for infinitely many orders: Welch (order p−1 for prime p) and Lempel–Golomb (order q−2, sometimes q−3, for prime powers q), plus variants. Complete enumeration by exhaustive search is known through order 29 (orders 28 and 29 have 712 and 164 arrays, counting rotations and reflections as distinct; OEIS A008404). The smallest orders for which no Costas array is known are 32 and 33 (Drakakis, "Open problems in Costas arrays").

What counts as progress

An explicit Costas array of order 32 or 33 (or any other order with no known array).
Complete enumeration of order 30 (or 31), with code, search-space partition and logs.
Proofs of non-existence for restricted families (e.g. arrays with a given symmetry or obtainable by a given extension of algebraic constructions), with the exhaustive search published.
Documented negative results for local-search or SAT approaches at order 32.

How it is checked — certificate format. A Costas array is submitted as one line of n integers, the permutation π(1), …, π(n) in 1..n. A short script checks it is a permutation and that all vectors (j − i, π(j) − π(i)) for i < j are distinct (O(n²) with a hash set). An enumeration claim ships the code, the partition of the search space into jobs, per-job counts, and the full list of arrays found, which a script re-checks individually and against the expected symmetry-class count.

Current state

The problem