A Costas array of order 32

A Costas array is a permutation matrix in which the vectors joining pairs of ones are all different, which gives it an optimal two-dimensional autocorrelation, useful in radar and sonar. Costas arrays are known for every order below 32, but nobody has found one of order 32 or proved that none exists. A candidate is checked in microseconds.

Problem

A Costas array of order $n$ is an $n \times n$ permutation matrix such that the $\binom{n}{2}$ vectors joining pairs of its ones are pairwise distinct. Equivalently, it is a permutation $f$ of $\{1, \dots, n\}$ such that for every shift $h$ the differences $f(i + h) - f(i)$ , over all $i$ with $1 \le i < i + h \le n$ , are pairwise distinct.

Question: does a Costas array of order 32 exist? It is the smallest order for which none is known.

Background

Costas introduced the arrays in 1965 to improve the detection performance of frequency-hopping sonar, and Gilbert studied them independently around the same time. Algebraic constructions over finite fields produce arrays for infinitely many orders: the Welch construction gives order $p - 1$ for every prime $p$ , the Golomb construction gives order $q - 2$ for every prime power $q$ , and variants remove corner dots to reach a few smaller orders. Other orders depend on exhaustive computer searches. Warnke, Correll and Swanson (2023) proved that Costas arrays become exponentially rare among permutations as $n$ grows.

What is known

Exhaustive enumeration is complete for all $n \le 29$ ; the searches for orders 28 and 29 took the equivalent of 70 and 366.55 years of single-CPU time (Drakakis, Iorio, Rickard and Walsh, 2011). Counts for $n \ge 30$ in Beard's database (2017) are only lower bounds.
Order 32: Drakakis's 2011 list of open problems gives it as the smallest order with no known Costas array, and Warnke, Correll and Swanson (2023) note that it remains unknown whether one exists. Each increase of the order needs roughly five times more computation for exhaustive search.
No nonexistence result is known for any order.

What would count as progress

A Costas array of order 32, given as a permutation. Checking the Costas property means comparing the $\binom{n}{2}$ displacement vectors, a few microseconds.
A new algebraic or combinatorial construction that reaches order 32, or more generally fills orders the known families miss.
A proof that no Costas array of order 32 exists, for example a search with symmetry reductions whose completeness can be audited; this would be the first nonexistence result for any order.

References

S. W. Golomb and H. Taylor, "Constructions and properties of Costas arrays", Proceedings of the IEEE 72 (1984), 1143–1163. doi:10.1109/proc.1984.12994
K. Drakakis, "A review of Costas arrays", Journal of Applied Mathematics (2006). doi:10.1155/jam/2006/26385
K. Drakakis, F. Iorio, S. Rickard and J. Walsh, "Results of the enumeration of Costas arrays of order 29", Advances in Mathematics of Communications 5 (2011), 547–553. doi:10.3934/amc.2011.5.547
L. Warnke, B. Correll and C. N. Swanson, "The density of Costas arrays decays exponentially", IEEE Transactions on Information Theory 69 (2023), 575–581. doi:10.1109/tit.2022.3202507

Source

Imported from Open problems in Costas arrays (K. Drakakis), curated by Konstantinos Drakakis. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.