Researchers Reveal Faster Enumeration of Hadamard Matrices up to Order
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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 25,000 for matrices of order 20. The study extends beyond enumeration, constructing quaternionic Hadamard matrices with potential applications in quantum communication and rigorously proving their non-equivalence to matrices generated by alternative methods. Furthermore, the researchers analytically investigated the properties of these matrices, demonstrating the feasibility of characterising them based on a fixed pattern of entries. These findings suggest a richer set of properties and indicate the potential for an abundance of quaternionic Hadamard matrices as their order increases. Experiments show that a finite sequence is considered perfect if it exhibits zero periodic autocorrelation after a nontrivial cyclic shift, a concept central to the construction of these matrices. The work establishes a direct link between perfect sequences and circulant Hadamard matrices, where the sequence defines the first row of the matrix.
This research not only advances the theoretical understanding of Hadamard matrices but also opens avenues for exploring their practical applications in fields reliant on efficient data encoding and signal processing. Enumeration of Williamson-type Hadamard matrices via asymmetric sequence analysis and block amicability provides new insights Scientists investigated quaternionic perfect sequences and their connection to Williamson-type Hadamard matrices, developing a novel enumeration algorithm to accelerate the search for these matrices. The research team devised an algorithm significantly faster than previous methods, crucially removing the requirement for input sequences to be symmetric. Implementing this algorithm, they successfully enumerated all circulant and potentially non-symmetric Williamson-type matrices of orders up to 21, exceeding the previously established limit of order 13. To enhance computational efficiency, the study pioneered a method leveraging the pairwise amicability of blocks within quaternion-type Hadamard matrices. This innovation dramatically reduced the number of block pairs requiring consideration, achieving a factor of over 25,000 in order 20. Researchers then constructed quaternionic Hadamard matrices, verifying their novelty by demonstrating non-equivalence to previously known matrices.
The team analytically studied the properties of these matrices, establishing the feasibility of characterizing them using a fixed pattern of entries. Experiments employed circulant block matrices, proving that pairwise amicability is equivalent to the conditions defining Williamson-type matrices under this configuration. This equivalence establishes a direct correspondence between Williamson-type sequences and QT sequences, formally defined in terms of correlation. The system delivers a robust framework for identifying and constructing these matrices, building upon prior work that identified lengths for which circulant symmetric Williamson-type matrices do not exist. Scientists harnessed the properties of quaternions, an extension of the complex plane with additional numbers i, j, and k, to define perfect sequences and establish their relationship to Hadamard matrix construction. This approach enables the discovery of perfect quaternion sequences of unbounded lengths, extending results previously limited to complex sequences. Enumeration of quaternionic perfect sequences and amicable block optimisation represent a novel approach Scientists have achieved a significant breakthrough in the enumeration of quaternionic perfect sequences, establishing a new computational milestone. Their work details an enumeration algorithm demonstrably faster than previous methods, crucially without requiring sequence symmetry. Implementing this algorithm, the team successfully enumerated all circulant and potentially non-symmetric Williamson-type matrices of orders up to 21, exceeding the previously established limit of order 13. This represents a substantial advancement in the field of Hadamard matrix construction. Experiments revealed that when the blocks of a quaternion-type Hadamard matrix are circulant, these blocks are necessarily pairwise amicable. This discovery dramatically improves the efficiency of the algorithm, reducing the number of block pairs needing consideration in order 20 by a factor exceeding 25,000. Measurements confirm this optimization significantly accelerates the process of identifying valid matrix configurations.
The team then leveraged these results to construct quaternionic Hadamard matrices, verifying their uniqueness compared to matrices generated by alternative methods. Further analytical study demonstrated the feasibility of characterizing quaternionic Hadamard matrices with a fixed pattern of entries. Data shows a richer set of properties exists within these matrices, suggesting an abundance of quaternionic Hadamard matrices for sufficiently large orders. The breakthrough delivers a powerful new tool for exploring and constructing these complex mathematical objects, with potential applications in areas like quantum communication. Tests prove the refined algorithm and analytical insights provide a robust foundation for future research into Hadamard matrices and their properties. Quaternion sequences facilitate Hadamard matrix enumeration up to order 21= Scientists have enumerated perfect quaternion sequences over the quaternions, Q+, for lengths up to 21. This work establishes a one-to-one correspondence between these sequences and binary sequences used in Williamson’s construction of quaternion-type Hadamard matrices, enabling a significantly faster enumeration algorithm than previously available. The algorithm does not require symmetry in the sequences, allowing for the exhaustive enumeration of matrices up to order 21, exceeding the previous limit of order 13. Researchers proved that when blocks within a quaternion-type Hadamard matrix are circulant, these blocks are necessarily pairwise amicable, substantially improving the efficiency of the enumeration process by reducing the number of block pairs needing consideration. This allowed the construction of new quaternion-type Hadamard matrices and confirmation that they are not equivalent to previously known matrices. Analytical study of these matrices demonstrated the feasibility of characterising them with a fixed pattern of entries, suggesting a richer set of properties and a potential abundance of such matrices at larger orders. The authors acknowledge a limitation in that their enumeration is currently restricted to orders up to 21. Future research could extend this enumeration to even larger orders, potentially confirming the conjecture that the multiplicity of equivalence classes of these matrices continues to increase. These findings contribute to a better understanding of quaternionic Hadamard matrices, which have applications in areas such as quantum communication protocols, and demonstrate the existence of a greater number of these matrices than previously recognised. 👉 More information 🗞 Quaternionic Perfect Sequences and Hadamard Matrices 🧠 ArXiv: https://arxiv.org/abs/2601.22337 Tags: