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A new approach to the excess problem of Hadamard matrices. (English) Zbl 1401.05054
Summary: In this paper, we give a new technique to find families of Hadamard matrices with maximum excess. In particular, we find regular or biregular Hadamard matrices with maximum excess by negating some rows and columns of known Hadamard matrices obtained from quadratic residues of finite fields. More precisely, we show that if either $$(2m+1)^2+2$$ or $$m^2+(m+1)^2$$ is a prime power, then there exists a biregular Hadamard matrix of order $$n=(2m+1)^2+3$$ with maximum excess. Furthermore, we give a sufficient condition for Hadamard matrices obtained from quadratic residues being transformed to regular ones in terms of four-class translation association schemes on finite fields. The core part of this paper is how to find “switching” sets of rows and columns.

##### MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 05B05 Combinatorial aspects of block designs 05E30 Association schemes, strongly regular graphs 11T22 Cyclotomy 11T24 Other character sums and Gauss sums
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