Two New Classes of Difference Families

Journal of Combinatorial Theory, Series A 90, 353355 (2000) doi:10.1006jcta.1999.3038, available online at http:www.idealibrary.com on

NOTE Two New Classes of Difference Families Marco Buratti Dipartimento di Ingegneria Elettrica, Universita de L'Aquila, Poggio di Roio, I-67040 L 'Aquila, Italy E-mail: burattimat.uniroma1.it Communicated by the Managing Editors Received January 23, 1999

We construct, in a very simple way, two new classes of elementary abelian (q 2, k, k&1) and (q 2, k+1, k+1) difference families with k a multiple of q&1. The first of these classes contains, as special cases, the supplementary difference systems constructed by A. Chaderjian (1997, J. Combin. Theory Ser. A 80, 218231).  2000 Academic Press

We refer to [1, 2] for general background on difference families (DFs). In [3], using the terminology of supplementary difference systems, the author essentially proves, in a quite involved way, that for q a prime power#3 (mod 4) and : a primitive root in the field F q 2 , the set D= q+1 [D 0 , D 1 , D 2 , D 3 ] with D i =[: h+(q+1)(i+4j )4 | 0h< 4 ; 0 j
2

2

2

2

353 0097-316500 35.00 Copyright  2000 by Academic Press All rights of reproduction in any form reserved.

354

NOTE

In the following q is a prime power and V is a 2-dimensional vector space over the finite field F q . If A is a subset of V, then A&[0] is denoted by A*. A m-cone of V is any subset of V which is union of m distinct 1-dimensional subspaces. If C is a m-cone, then C* is a m*-cone. Theorem. Let q+1=mn and let F=[C 1 , ..., C n ] be a set of m-cones of V such that the corresponding set of m*-cones F*=[C * 1 , ..., C *] n partitions V*. Then F and F* respectively are (q 2, k+1, k+1) and (q 2, k, k&1) difference families where k=m(q&1). Proof. Let C be a m-cone. By definition, C*= m i=1 S i* where the S i 's are pairwise distinct 1-dimensional subspaces of V. We have *) _ . 2S *= . [V*&(S * _ . (q&2) S * 2C*= . (S *&S i j i i _ S *)] j i . i{ j

i

i{ j

i

2

It easily follows that 2C*=(m &m) V* _ (q&2m) C*. Now, let F=[C 1 , ..., C n ] be a set of m-cones such that F*= [C * partitions V*. We have 1 , ..., C *] n 2 &m) nV* _ (q&2m) . C * 2F*=. 2C *=(m i i i

i

=(m&1)(q+1) V* _ (q&2m) V*=[m(q&1)&1] V*. 2F=. 2C i =. (2C * _ 2V*=[m(q&1)+1] V*. i _ \C *)=2F* i i

i

The assertion follows. K (q+1) !

Note that we have exactly n! (m!) ways of applying the above theorem since this is just the number of distinct partititons of V* into m*-cones. It should be interesting to establish how many of the corresponding DFs are pairwise inequivalent. Now, observing that each 1-dimensional subspace of V is a coset of (q+1)th powers in F q 2 and zero, it is easy to see that the difference families D and E mentioned at the beginning of this note are sets of ( q+1 4 )*-cones partitioning V*. It follows that they are obtainable applying the above theorem with m= q+1 4 . n

REFERENCES 1. R. J. R. Abel, Difference families, in ``CRC Handboock of Combinatorial Designs'' (C. J. Colbourn and J. H. Dinitz, Eds.), pp. 270287, CRC Press, Boca Raton, FL, 1996.

NOTE

355

2. T. Beth, D. Jungnickel, and H. Lenz, ``Design Theory,'' Cambridge Univ. Press, Cambridge, 1995. 3. A. Chaderjian, New classes of residue supplementary difference systems, J. Combin. Theory Ser. A 80 (1997), 218231. 4. R. M. Wilson, Cyclotomy and difference families in elementary abelian groups, J. Number Theory 4 (1972), 1747.