A survey of partial difference sets

Abstract

Abstract. Let G be a finite group of order v. A k-element subset D of G is called a (v, k, 3,,/z)-partial difference set if the expressions gh -1, for g and h in D with g ~ h, represent each nonidentity element in D exactly ), times and each nonidentity element not in D exactly # times. If e t~ D and g E D iff g-1 E D, then D is essentially the same as a strongly regular Cayley graph. In this survey, we try to list all important existence and nonexistence results concerning partial difference sets. In particular, various construction methods are studied, e.g., constructions using partial congruence partitions, quadratic forms, cyclotomic classes and finite local rings. Also, the relations with Schur tings, two-weight codes, projective sets, difference sets, divisible difference sets and partial geometries are discussed in detail. Strongly Regular Graphs and Partial Difference SetsA graph I TM with v vertices is said to be a (v, k, k, #)-strongly regular graph if (i) it is regular of valency k, i.e., each vertex is joined to exactly k other vertices; (ii) any two adjacent vertices are both joined to exactly 3, other vertices and two nonadjacent vertices are both joined to exactly/z other vertices. A Cayley graph is defined as a graph r = (v, E) which admits an automorphism group G acting regularly on the vertex set V (see Yap [7] for background on Cayley graphs). If we identify the vertices of I ~ with the elements of the regular automorphism group G, then r can be generated by a subset D of G such that two vertices g, h E G are joined nonabelian) if the group G is abelian (resp. nonabelian). For a subset S of a group G, let S it) = {g~ : g E S} for any integer t.

QUASI m-CAYLEY STRONGLY REGULAR GRAPHS

“…, C e−1 , where C i = θ i G and θ is a primitive root of F q . In many cases, a partial difference set D can be found by taking an appropriate union of orbits (see [8]), and in this case the strongly regular graph generated by D is obviously a quasi m-Cayley graph, where the point at infinity is the zero element of the field.…”

“…However, among the other 14 graphs in this family there are six more graphs that are quasi m-Cayley graphs on some cyclic group. In particular, two of the graphs are quasi m-Cayley graphs on a cyclic group C n for each (m, n) ∈ {(4, 6), (6,4), (8,3)}, and four of the graphs are quasi 8-Cayley graphs on a cyclic group C 3 (but are neither quasi 4-Cayley graphs nor 6-Cayley graphs on a cyclic group). The symbols of the graphs that are distinct from the Paley graph P (25) are listed in the appendix (only the symbols with respect to a quasi (4, 6)-semiregular automorphism are given for the two first ones).…”

“…For example, one can ask when there exist a group of automorphisms G of the graph acting regularly on its vertices, that is, acting transitively on the vertex set with trivial vertex stabilizers (in other words, when a SRG G is a Cayley graph). It is known that this is the case if there exists a (v, k, λ, µ)-partial difference set D in G (see for example [8]); this is a subset D of G, with |G| = v and |D| = k such that 0 ∈ D, D = −D and every element in D (respectively, every non-zero element in G − D) can be expressed in exactly λ ways (respectively, in exactly µ ways) as a difference of two distinct elements of D. The (v, k, λ, µ)-SRG generated by D is the graph with vertex set G where xy is an edge if y − x ∈ D. If we relax the condition that G acts transitively but keep the condition that all the stabilizers are trivial, then we are asking for SRGs G admitting a group of automorphisms G acting semiregularly on the vertex set of G. This question was first studied by Marušič [12] and by de Resmini and Jungnickel [13]. In particular, they studied SRGs admitting a semiregular group of automorphisms with two orbits on the vertex set, and proved that the existence of such graphs is equivalent to the existence of certain algebraic structures that de Resmini and Jungnickel termed partial difference triples and that were specially studied by Leung and Ma [7].…”

Partial difference sets and amorphic group schemes from pseudo-quadratic bent functions

“…, R d ) a group scheme if X is a group and (xg, yg) ∈ R i whenever (x, y) ∈ R i for all g ∈ X and all i = 0, . Such a subring is also known as a Schur ring of G [29] and it is the Bose-Mesner ring of the scheme. Conversely, every Schur ring of G gives rise to a group scheme of G and such a scheme will be simply denoted by (G; D 0 , D 1 , .…”

“…partial difference sets containing the identity of G, can be similarly defined. The reader can find more details on strongly regular graphs in [6,19] and on partial difference sets in [29].…”