A note on supplementary difference sets

Let S1, S2,···, Sn be subsets of G, a finite abelian group of order v, containing k1, k2,...,kn elements respectively. Write Ti for the totality of all differences between elements of Si (with repetitions), and T for the totality of elements of all the Ti. We will denote this by T= T1 & T2 & ... & Tn. If T contains each non-zero element of G a fixed number of times, lambda say, then the sets S1, S2, ..., Sn will be called n-{v; k1, k2, ..., kn ; lambda} supplementary difference sets. Disciplines Physical Sciences and Mathematics Publication Details Jennifer Seberry Wallis, A note on supplementary difference sets, Aequationes Mathematicae, 10, (1974), 46-49. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/957 Vol. 10, fase. 1, 1974 Reprint from aequationes mathematicae BIRKHAUSER VERLAG BASEL A Note on Supplementary Difference Sets JENNIFER WALLIS (Newcastle, New South Wales, Australia) pages 46-49 Let S1' S2,···, Sn be subsets of G, a finite abelian group of order v, containing kl> k 2 , ••• , kn elements respectively. Write Ti for the totality of all differences between elements of Si (with repetitions), and T for the totality of elements of all the T i• We will denote this by T= T1 & T2 & ... & Tn. If T contains each non-zero element of G a fixed number of times, A say, then the sets S1' S2, ... ' Sn will be called n-{v; k1' k2' ... , k n ; A} supplementary difference sets. If k1 = k2 = ... = kn = k we will write 11{v; k; A} to denote the supplementary difference sets. If k1 =k2 = ... =ki' k i+1 =ki+2 = ... =ki+ j' ... , k z= ... =kn then sometimes we write 11-{V; i:k1,j:ki+1' ... ; A}. It can be easily seen by counting the differences that the parameters of 11{v; k1' k 2, ... , kn; ).} supplementary difference sets satisfy A(v 1) = L kj(kj -1). j= 1 We use braces, { }, to denote sets and square brackets, [ ], to denote collections where repetitions may remain. We now let v = 4r (2,1, + 1) + 1 = p Y, where p is a prime and further let H { 4r j + i . 0 --. --2'} i = X • ""'-J""'/, , i = 0, 1, ... , 4r 1 with x a primitive element of GF( v). Write for some m, O<m <2r, where the i j are distinct integers. Now we consider the differences between elements of H 2 ;, that is, the collection [X4rj+2i _ x4rZ+2i:j # 1,0 ,;;;j, 1,;;; 2,1,] (1) = {x4rj+2i:0';;;j';;; 2A} times [1 x4r(Z-j): 1 #j, 0,;;; I,;;; 21.J =H2i times [l-x 4r(l-j):I#j,0,;;;1,;;;2A] and, since any element of a group multiplied onto a coset gives a coset, this expression must represent co sets with certain multiplicities, say bk, write