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The cocyclic Hadamard matrices of order less than 40

机译：小于40的Cocyclic Hadamard矩阵

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In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and 36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey, Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4t, 2, 4t, 2t)-relative difference sets in the groups of orders 64 and 72.
机译：本文对所有小于40的共循环Hadamard矩阵进行了分类。也就是说，所有此类Hadamard矩阵都是明确构造的，直到Hadamard等价。这是de Launey和Ito的工作的重大扩展和完成。讨论了同周期发展的理论，并描述了确定给定Hadamard矩阵是否为同周期的算法。由于已对所有最多28个阶的Hadamard矩阵进行了分类，因此该算法足以对最多28个阶的共循环Hadamard矩阵进行分类。甚至不知道32和36阶Hadamard矩阵的总数。因此，我们使用一种不同的方法来按这些顺序构造所有共循环Hadamard矩阵。 de Launey，Flannery和Horadam对共循环Hadamard矩阵与相对差集之间关系的结果用于对32和36阶的共循环Hadamard矩阵进行分类。这是通过对（4t，2，订单组64和72中的4t，2t）相对差集。

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