Researchers Reveal Faster Enumeration of Hadamard Matrices up to Order
Researchers are investigating the properties of perfect sequences, finite sequences with unique autocorrelation characteristics, and their connection to the construction of Hadamard matrices, mathematical objects crucial in signal processing and quantum computing. Aidan Bennett from the University of Windsor, Curtis Bright and Ashwin Nayak from the University of Waterloo, alongside Paul Colinot from Université Grenoble Alpes, present a novel enumeration algorithm for quaternionic perfect sequences, significantly accelerating the search for these structures and bypassing limitations of previous methods. Their work extends exhaustive enumeration to orders up to 21, exceeding the prior limit of 13, and establishes key relationships between the building blocks of quaternion-type Hadamard matrices, dramatically improving computational efficiency. This advancement not only facilitates the construction of new Hadamard matrices but also suggests the potential for a more comprehensive understanding and characterisation of these complex mathematical entities at larger scales. The research focuses on quaternionic perfect sequences, which exhibit a one-to-one correspondence with binary sequences used in Williamson’s construction of quaternion-type Hadamard matrices. By leveraging this connection, the team devised a novel enumeration algorithm that surpasses the speed of previous methods and crucially, does not require the sequences to be symmetric. Implementing this algorithm, researchers successfully enumerated all circulant and potentially non-symmetric Williamson-type matrices of orders up to 21, a substantial leap from the previously exhaustively enumerated order of 13. This achievement was facilitated by a key discovery: when the blocks of a quaternion-type Hadamard matrix are circulant, those blocks are necessarily pairwise amicable. This property dramatically streamlined the filtering process, reducing the number of block pairs needing consideration by a factor exceeding
