On amicable sequences and orthogonal designs
This conference was originally published as Georgiou, S, Koukouvinos, C and Seberry, J, On amicable sequences and orthogonal designs, Proceedings of the 23rd Australasian Conference on Combinatorial Mathematics and Combinatorial Computing, Christchurch, New Zealand, 4-7 December 2000.
In this paper we give a general theorem which can be used to multiply the length of amicable sequences keeping the amicability property and the type of the sequences. As a consequence we have that if there exist two, four or eight amicable sequences of length m and type (al, a2), (al, a2, a3, a4) or (al, a2, ... , a8) then there exist amicable sequences of length ℓ ≡ 0 (mod m) and of the same type. We also present a theorem that produces a set of 2v amicable sequences from a set of v (not necessary amicable) sequences and a construction method for amicable sequences of type (al, al, a2, a2, ... , av, av) from v pairs of disjoint (0, ±1) amicable sequences.
Using these results we can obtain many infinite classes of orthogonal designs. Actually, if there exists an orthogonal design of order n and of type (al, a2,... , av), which is constructed from sequences, then there exists an infinite family of orthogonal designs of the same type which is constructed from appropriate sequences.
S. Georgiou, C. Koukouvinos, and J. Seberry. "On amicable sequences and orthogonal designs" Faculty of Informatics - Papers.. Dec. 2000.
Available at: http://works.bepress.com/jseberry/98