An experimental search and new combinatorial designs via a generalisation of cyclotomy

Cyclotomy can be used to construct a variety of combinatorial designs, for example, supplementary difference sets, weighing matrices and T -matrices. These designs may be obtained by using linear combinations of the incidence matrices of the cyclotomic cosets. However, cyclotomy only works in the prime and prime power cases. We present a generalisation of cyclotomy and introduce generalised cosets. Combinatorial designs can now be obtained by a search through all linear combinations of the incidence matrices of the generalised cosets. We believe that this search method is new. The generalisation works for all cases and is not restricted to prime powers. The paper presents some new combinatorial designs. We give a new construction for T -matrices of order 87 and hence an OD(4 x 87;87,87,87,87). We also give some D-optimal designs of order n = 2v = 2 x 145,2 x 157,2 x 181.