Quaternionic Perfect Sequences and Hadamard Matrices

2026-01-29

AI Summary

This study investigates the construction of circulant Williamson-type quaternion Hadamard matrices, with a focus on the efficient enumeration of asymmetric quaternion perfect sequences. By establishing a one-to-one correspondence with binary Williamson sequences, the work proposes a fast enumeration algorithm that dispenses with symmetry assumptions and proves that circulant blocks must be pairwise amicable, substantially enhancing search efficiency. Leveraging analyses of periodic autocorrelation, quaternion algebra, and matrix equivalence, the approach extends the feasible enumeration order from 13 to 21. At order 20, the search space is reduced by over 25,000-fold, enabling the construction of novel, non-equivalent quaternion Hadamard matrices with potential applications in quantum communication.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Applications of cryptography

Abstract

A finite sequence of numbers is perfect if it has zero periodic autocorrelation after a nontrivial cyclic shift. In this work, we study quaternionic perfect sequences having a one-to-one correspondence with the binary sequences arising in Williamson's construction of quaternion-type Hadamard matrices. Using this correspondence, we devise an enumeration algorithm that is significantly faster than previously used algorithms and does not require the sequences to be symmetric. We implement our algorithm and use it to enumerate all circulant and possibly non-symmetric Williamson-type matrices of orders up to 21; previously, the largest order exhaustively enumerated was 13. We prove that when the blocks of a quaternion-type Hadamard matrix are circulant, the blocks are necessarily pairwise amicable. This dramatically improves the filtering power of our algorithm: in order 20, the number of block pairs needing consideration is reduced by a factor of over 25,000. We use our results to construct quaternionic Hadamard matrices of interest in quantum communication and prove they are not equivalent to those constructed by other means. We also study the properties of quaternionic Hadamard matrices analytically, and demonstrate the feasibility of characterizing quaternionic Hadamard matrices with a fixed pattern of entries. These results indicate a richer set of properties and suggest an abundance of quaternionic Hadamard matrices for sufficiently large orders.

Problem

• quaternionic perfect sequences
• Hadamard matrices
• Williamson construction
• circulant matrices
• periodic autocorrelation

Innovation

• quaternionic Hadamard matrices
• perfect sequences
• Williamson construction
• circulant blocks
• pairwise amicable