Short Amicable Sets

Abstract: A pair of matrices X and Y are said to be amicable if XYT = YXT. In this paper, if X and Y are orthogonal designs, group generated or circulant on the group G, these will be denoted 2—SAS(n; ul, u2; G). Recently Kharaghani, in "Arrays for orthogonal designs", J. Combin. Designs, 8 (2000), 166-173, extended this concept to an amicable set, {Ai}2n, i=1, of 2n circulant matrices, which satisfy ∑(Aσ(2i-1)AT σ(2i-1) - Aσ(2i) AT (2i-1) = 0. In this paper we concentrate on constructing short amicable sets, which satisfy the same equation but contain four, called short, or two, called 2-short, matrices. We give a method of multiplying the order of 2-short circulant amicable sets and thus we obtain many infinite classes of 2-short circulant amicable sets. We give some constructions for infinite families of circulant amicable sets. We then contrast by comparing with short block amicable sets which are block circulant matrices and defined on a group G1 x G2.