Primitive Symmetric Designs with Prime Power Number of Points Snje ˇ zana Brai ´ c, Anka Golemac, Joˇ sko Mandi ´ c, Tanja Vu ˇ ci ˇ ci ´ c Department of Mathematics, University of Split, Teslina 12/III, 21 000 Split, Croatia Received December 23, 2008; revised July 26, 2009 Published online 12 October 2009 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.20240 Abstract: In this paper we either prove the non-existence or give explicit construction of primitive symmetric (v, k, k) designs with v = p m <2500, p prime and m>1. The method of design construction is based on an automorphism group action; non-existence results additionally include the theory of difference sets, multiplier theorems in particular. The research involves programming and wide-range computations. We make use of software package GAP and the library of primitive groups which it contains. 2009 Wiley Periodicals, Inc. J Combin Designs 18: 141–154, 2010 Keywords: symmetric design; primitive automorphism group; difference set 1. INTRODUCTION AND PRELIMINARIES In geometrical sense, symmetric designs with transitive automorphism groups are considerably rich structures. Primitive symmetric designs, meaning symmetric designs with automorphism groups acting primitively on the point set, form one subclass of “transitive” designs. It is known that symmetric designs with 2-transitive automorphism groups are examples of primitive designs [10]. These designs are called Kantor designs. Another example is provided by Paley difference sets; they yield a series of primitive symmetric designs called Paley designs. This research considers symmetric (v, k , ) designs which admit a point-primitive group of automorphisms of prime power degree p m <2500, m>1. Up to a few unde- cided cases, all designs with these properties are classified. We confine ourselves to v = p m >2k , because the complement of a primitive design is also primitive with the same full automorphism group. The number of obtained designs is 61. Their parameters are either that of Paley (5 triples) or of Kantor designs (5 triples); we call these parameter triples Paley type and Kantor type, respectively. For Paley-type parameters (243, 121, 60), (343, 171, 85), and (1331, 665, 332) we have constructed 26 primitive symmetric designs which are Journal of Combinatorial Designs 2009 Wiley Periodicals, Inc. 141