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On the excess of Hadamard matrices. (English) Zbl 0667.05013
Bounds for the excess \(\sigma\) (H) of a Hadamard matrix H the sum of the elements of the matrix H, are established in the work. \(\sigma\) (n) denotes the maximum of \(\sigma\) (H) for all Hadamard matrices of order n. Upper bounds for \(\sigma\) (n), with n from various intervals, are obtained, improving the well-known bound \(n\sqrt{n}\). The established bound coincides with \(n\sqrt{n}\) when \(n=(4m)^ 2\) or \(n=4(2m+1)^ 2\) and the matrix H is regular, i.e. it has all row and column sums equal.
Regular matrices are constructed for \(n=4(2m+1)^ 2\) when \(m=1,2,3\) and their construction for \(m=4,9,12\) is indicated. A procedure is described to find all row-sum and column-sum vectors of a Hadamard matrix with given excess.
Reviewer: S.S.Agayan

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
Full Text: DOI
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