ABSTRACT Through the use of regularizing vectors, all regular quaternary Hadamard matrices of orders 10 and 18 have been successfully identified. Of these, two matrices of order 10 and 184 matrices of order 18 were found to have unbiased mates. Converting the quaternary Hadamard matrices of order 18 to real Hadamard matrices, the study uncovered that six matrices of order 36 having unbiased pairs were connected to the extremal Pless symmetry code. Furthermore, the study uncovered exactly 28 nonregular quaternary Hadamard matrices of order 18, providing the first examples of nonregular Hadamard matrices of order 36 and, as a result, settling a recent conjecture in the negative.
Kharaghani et al. (2026) studied this question.
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