Geodesic
Skew-Hadamard Matrices of the Goethals-Seidel Type
Canadian journal of mathematics, Tome 27 (1975) no. 3, pp. 555-560

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1. Introduction. We prove, using a theorem of M. Hall on cyclic projective planes, that if g is a prime power such that either 1 + q + q2 is a prime congruent to 3, 5 or 7 (mod 8) or 3 + 2q + 2q2 is a prime power, then there exists a skew-Hadamard matrix of the Goethals-Seidel type of order 4(1 + q + q2). (A Hadamard matrix H is said to be of skew type if one of H + I, H — lis skew symmetric. ) If 1 + q + q2 is a prime congruent to 1 (mod 8), then a Hadamard matrix, not necessarily of skew type, of order 4(1 + q + q2) is constructed. The smallest new Hadamard matrix obtained has order 292. 
Spence, Edward. Skew-Hadamard Matrices of the Goethals-Seidel Type. Canadian journal of mathematics, Tome 27 (1975) no. 3, pp. 555-560. doi: 10.4153/CJM-1975-066-9
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