The Construction of Regular Hadamard Matrices by Cyclotomic Classes

© 2020, Iranian Mathematical Society. For every prime power q≡7mod16, there are (q; a, b, c, d)-partitions of GF(q), with odd integers a, b, c, and d, where a≡±1mod8 such that q= a2+ 2 (b2+ c2+ d2) and d2= b2+ 2 ac+ 2 bd. Many results for the existence of 4-{q2;q(q-1)2;q(q-2)} SDS which 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 SDS becomes 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