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Embedding cocylic D-optimal designs in cocylic Hadamard matrices

Victor Alvarez, Jose Andres Armario, Maria Dolores Frau, Felix Guidiel
2012 The Electronic Journal of Linear Algebra  
A method for embedding cocyclic submatrices with "large" determinants of orders 2t in certain cocyclic Hadamard matrices of orders 4t is described (t an odd integer). If these determinants attain the largest possible value, we are embedding D-optimal designs. Applications to the pivot values that appear when Gaussian elimination with complete pivoting is performed on these cocyclic Hadamard matrices are studied. In 1969, Cryer [7] conjectured that if A is a
more » ... d that if A is a real n × n matrix such that |a i,j | ≤ 1, then g(n, A) ≤ n, with equality if and only if A is a Hadamard matrix. In 1991 Gould [15] proved that the first part of the conjecture is not true. He found matrices with growth bigger than their orders. Thus, the following remains open: Conjecture(Cryer) The growth of a Hadamard matrix is its order. This conjecture has been proven only for n = 4, 8, 12 and 16 (see [7, 10, 23] ).
doi:10.13001/1081-3810.1580 fatcat:kwvybo2ug5gendly4wyo373fvy