Contract grant sponsor: National Science Foundation of China; Contract grant numbers: 61070215, 61103191.
(2n,2n,2n,1)-Relative Difference Sets and Their Representations
Article first published online: 16 APR 2013
© 2013 Wiley Periodicals, Inc.
Journal of Combinatorial Designs
Volume 21, Issue 12, pages 563–584, December 2013
How to Cite
Zhou, Y. (2013), (2n,2n,2n,1)-Relative Difference Sets and Their Representations. J. Combin. Designs, 21: 563–584. doi: 10.1002/jcd.21349
- Issue published online: 2 OCT 2013
- Article first published online: 16 APR 2013
- Manuscript Revised: 21 MAR 2013
- Manuscript Received: 5 JUN 2012
- National Science Foundation of China. Grant Numbers: 61070215, 61103191
- relative difference set;
- commutative semifield;
- projective plane;
We show that every -relative difference set D in relative to can be represented by a polynomial , where is a permutation for each nonzero a. We call such an f a planar function on . The projective plane Π obtained from D in the way of M. J. Ganley and E. Spence (J Combin Theory Ser A, 19(2) (1975), 134–153) is coordinatized, and we obtain necessary and sufficient conditions of Π to be a presemifield plane. We also prove that a function f on with exactly two elements in its image set and is planar, if and only if, for any .