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Circulant 16-Modular Hadamard Matrices and Jacobi Sums

✍ Scribed by Shalom Eliahou; Michel Kervaire

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 190 KB
Volume: 100
Category: Article
ISSN: 0097-3165
DOI: 10.1006/jcta.2002.3283

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✦ Synopsis

We are concerned here with the existence problem of 16-modular circulant Hadamard matrices H of size 4p (p prime), satisfying the additional condition that any two rows at distance n=2 in H are strictly orthogonal. A necessary existence condition is p 1 mod 8: For p 1 mod 16; existence follows from the more general result of a previous paper of ours, showing the existence of ðp À 1Þ-modular matrices of the above kind. In the remaining case p 9 mod 16; we construct explicit examples which solve the problem whenever 2 is a fourth power mod p: When 2 is not a fourth power mod p; we conjecture that such matrices cannot exist.

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