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Hadamard matrices, finite sequences, and polynomials defined on the unit circle


Author: C. H. Yang
Journal: Math. Comp. 33 (1979), 688-693
MSC: Primary 05B20; Secondary 15A57
DOI: https://doi.org/10.1090/S0025-5718-1979-0525685-8
MathSciNet review: 525685
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Abstract: If a $ (\ast)$-type Hadamard matrix of order 2n (i.e. a pair (A, B) of $ n \times n$ circulant (1,-1) matrices satisfying $ AA\prime + BB\prime = 2nI$) exists and a pair of Golay complementary sequences (or equivalently, two-symbol $ \delta $-code) of length m exists, then a $ (\ast)$-type Hadamard matrix of order 2mn also exists. If a Williamson matrix of order 4n (i.e. a quadruple (W, X, Y, Z) of $ n \times n$ symmetric circulant (1,-1) matrices satisfying $ {W^2} + {X^2} + {Y^2} + {Z^2} = 4nI$) exists and a four-symbol $ \delta $-code of length m exists, then a Goethals-Seidel matrix of order 4mn (i.e. a quadruple (A, B, C, D) of $ mn \times mn$ circulant (1, -1) matrices satisfying $ AA\prime + BB\prime + CC\prime + DD\prime = 4mnI$) also exists. Other related topics are also discussed.


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DOI: https://doi.org/10.1090/S0025-5718-1979-0525685-8
Article copyright: © Copyright 1979 American Mathematical Society

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