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

Yang, C. H.

doi:10.1090/S0025-5718-1979-0525685-8

Hadamard matrices, finite sequences, and polynomials defined on the unit circle
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Math. Comp. 33 (1979), 688-693 Request permission

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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