An Exhaustive Census of Main-Diagonal Symmetric Costas Arrays of Orders 37-42
Abstract
We determine by exhaustive enumeration all main-diagonal symmetric Costas arrays of orders 37 through 42. There are 4 arrays of order 37, none of order 38, 16 of order 39, 2 of order 40, 12 of order 41, and 4 of order 42. All belong to known finite-field constructions. The arrays at orders 37-40 are Lempel arrays over 𝔽41 or their corner deletions and augmentations; the twelve arrays at order 41 are the Lempel arrays over 𝔽43; and the order-42 census consists of two corner augmentations and a reverse-complement pair in the Rickard-Golomb family. Combined with the published census through order 36, these results show that no sporadic main-diagonal symmetric Costas array occurs at orders 24-42. We also develop a collision model for random involutions. The first two terms of its collision exponent are derived analytically, n2/18+n3/2/360, with an O(n) remainder. The lower-order model and clumping correction are empirical. The six new censuses are consistent with the model's prediction of a rapidly declining expected sporadic population.
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BibTeX
@misc{dumitru2026exhaustive,
title = {{An Exhaustive Census of Main-Diagonal Symmetric Costas Arrays of Orders 37-42}},
author = {Bogdan Dumitru and Cristian Budala},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28690},
eprint = {2608.28690},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28690},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Exhaustive Census of Main-Diagonal Symmetric Costas Arrays of Orders 37-42
AU - Bogdan Dumitru
AU - Cristian Budala
PY - 2026
DA - 2026/08/26
PB - arXiv
DO - 10.48550/arxiv.2608.28690
UR - https://arxiv.org/abs/2608.28690
AB - We determine by exhaustive enumeration all main-diagonal symmetric Costas arrays of orders 37 through 42. There are 4 arrays of order 37, none of order 38, 16 of order 39, 2 of order 40, 12 of order 41, and 4 of order 42. All belong to known finite-field constructions. The arrays at orders 37-40 are Lempel arrays over 𝔽_41 or their corner deletions and augmentations; the twelve arrays at order 41 are the Lempel arrays over 𝔽_43; and the order-42 census consists of two corner augmentations and a reverse-complement pair in the Rickard-Golomb family. Combined with the published census through order 36, these results show that no sporadic main-diagonal symmetric Costas array occurs at orders 24-42. We also develop a collision model for random involutions. The first two terms of its collision exponent are derived analytically, n^2/18+n^3/2/360, with an O(n) remainder. The lower-order model and clumping correction are empirical. The six new censuses are consistent with the model's prediction of a rapidly declining expected sporadic population.
ER -