New Partial Difference Sets and a Related Problem about Galois Rings
by
Xiang-dong Hou
Department of Math. and Stat., Wright State Univ., Dayton, OH

We generalize a construction of partial difference sets (PDS) by Chen, Ray-Chaudhuri and Xiang through a study of the Teichmuller sets of Galois rings. Let p be a prime and let R=GR(p², t) be the Galois ring of characteristic p² and rank t. Let T be the Teichmuller set of R and let \tau:R/pR --> T be the inverse of \pi|_T where \pi:R --> R/pR is the projection. Define s(p, t) to be the largest dimension of a subspace W subset R/pR such that \tau(W) generates a subgroup of R of rank < t. We give a construction of a PDS in R with the parameters v=p^2t, k=r(p^t-1), \lambda = p^t+r²-3r, \mu = r²-r, where r=lp^{t-s(p, t)}, 1 <= l <= p^{s(p, t)}. We prove two lower bound for the function s(p, t): (1) s(p, t) >= the largest proper divisor of t; (2) s(p, t) >= the largest integer s such that ((s+p-1) || p) < t. The first bound produces the result of Chen, Ray-Chaudhuri and Xiang; the second bound gives many PDS with new parameters.

Date received: November 16, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-25.