On Jacobi sums, difference sets and partial difference sets in Galois domains

Abstract

The problems studied in this paper resemble in spirit partially the following two theorems due to Stanton and Sprott (Can J Math 10:73---77, 1958) and Whiteman (Ill J Math 6:107---121, 1962) and are based on ideas given in Storer (Cyclotomy and difference sets. Lectures in advanced mathematics, 1967). Theorem 1 (Stanton, Sprott) Let g be a primitive root of both$$p$$pand$$p -2$$p-2, where$$p$$pand$$p-2$$p-2are a pair of twin primes. Then the numbers$$\begin{aligned} 1, g, g^2, \ldots g^{(p^2 -3)/2}\, together\, with \,0, p+2, , 2(p +2), \ldots , (p- 1)(p +2) \end{aligned}$$1,g,g2,źg(p2-3)/2togetherwith0,p+2,,2(p+2),ź,(p-1)(p+2)form a difference set modulo$$p(p+2)$$p(p+2)with parameters$$ v=p(p + 2), k= (v - 1)/2, \lambda =(v- 3)/4$$v=p(p+2),k=(v-1)/2,ź=(v-3)/4 Theorem 2 (Whiteman) Let$$p$$pand$$q$$qbe two primes such hat$$(p- 1, q- 1)= 4$$(p-1,q-1)=4, and let$$ d = (p- 1)(q -1)/4$$d=(p-1)(q-1)/4. Then there are exactly two subgroups$$U_1,U_2$$U1,U2generated by a common primitive root of$$p$$pand$$q$$q, and one (but not both) of he sets$$\begin{aligned}&U_1\, together\, with \,0, q, 2q,\ldots , (p - 1)q\\&U_2\, together\, with \,0, q, 2q,\ldots , (p- 1)q \end{aligned}$$U1togetherwith0,q,2q,ź,(p-1)qU2togetherwith0,q,2q,ź,(p-1)qis a difference set modulo$$p q $$pqwith parameters$$v = pq, k= (v- 1)/4, \lambda = (v -5)/16 $$v=pq,k=(v-1)/4,ź=(v-5)/16if and only if$$ q = 3p +2$$q=3p+2and$$(v- 1)/4$$(v-1)/4is an odd square. However, we do not follow the classical method of cyclotomic numbers to generalize these results but rather present in a modern setting the connection between difference sets or partial difference sets and Jacobi sums. With the help of Jacobi sums we are able to rephrase the existence of difference sets, partial difference sets and so called $$G$$G-sets in Galois domains into a fundamental equation on Jacobi sums presented in Theorem 14. As an application we determine all cyclotomic difference and partial difference sets of small order or which are semiprimitive modulo its order by restricting the theorem to a product of two finite fields.