The Construction of Regular Hadamard Matrices by Cyclotomic Classes

Authors

Tianbing Xia, University of WollongongFollow
Mingyuan XiaFollow
Jennifer Seberry, University of WollongongFollow

RIS ID

143974

Publication Details

Xia, T., Xia, M. & Seberry, J. (2020). The Construction of Regular Hadamard Matrices by Cyclotomic Classes. Bulletin of the Iranian Mathematical Society,

Abstract

© 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.