The Construction of Regular Hadamard Matrices by Cyclotomic Classes

摘要

For every prime power q = 7 mod 16, there are (q; a, b, c, d)-partitions of GF(q), with odd integers a, b, c, and d, where a = +/- 1 mod 8 such that q = a(2)+2(b(2)+c(2)+d(2)) and d(2) = b(2)+ 2ac+ 2bd. Many results for the existence of 4-{q(2); q(q-1)/2; q(q-2)} SDSwhich are simple homogeneous polynomials of parameters a, b, c and d of degree at most 2 have been found. Hence, for each value of q, the construction of SDSbecomes equivalent to building a (q; a, b, c, d)-partition. Once this is done, the verification of the construction only involves verifying simple conditions on a, b, c and d which can be done manually.

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