发表机构
University of Bucharest（布加勒斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文穷尽枚举得到37-42阶主对角线对称Costas阵列数量，结合已有结果表明24-42阶无零散此类阵列，还提出随机对合的碰撞模型且新普查结果与模型预测一致。
AI 中文摘要
本文通过穷尽枚举确定了37至42阶的所有主对角线对称Costas阵列：37阶有4个，38阶无，39阶有16个，40阶有2个，41阶有12个，42阶有4个。所有阵列均属于已知的有限域构造：37至40阶阵列是有限域$\boldsymbol{F}_{41}$上的Lempel阵列，或其角删除与扩充；41阶的12个阵列是$\boldsymbol{F}_{43}$上的Lempel阵列；42阶普查结果包含2个角扩充及Rickard-Golomb族中的一对逆补对。结合已发表的36阶以内的普查结果，这些结果表明24至42阶不存在零散的主对角线对称Costas阵列。本文还提出了一个针对随机对合的碰撞模型，解析推导了其碰撞指数的前两项：$n^2/18+n^{3/2}/360$，余项为$O(n)$，低阶模型与团块修正为经验性内容。新普查的6组结果与模型预测的零散种群快速衰减的结论一致。
英文摘要
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 $\mathbb{F}_{41}$ or their corner deletions and augmentations; the twelve arrays at order 41 are the Lempel arrays over $\mathbb{F}_{43}$; 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.
