Siu Lun Ma

A survey of partial difference sets

LetG be a finite group of order ν. Ak-element subsetD ofG is called a (ν,k, λ, μ)-partial difference set if the expressionsgh−1, forg andh inD withg≠h, represent each nonidentity element inD exactly λ times and each nonidentity element not inD exactly μ times. Ife∉D andg∈D iffg−1∈D, thenD 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 rings, two-weight codes, projective sets, difference sets, divisible difference sets and partial geometries are discussed in detail.

Partial difference sets

A (?,k,?,β)-partial difference set in a finite group G of order ? is a subset D of G with k distinct elements such that expressions dnd?12 for d1 and d2 in D, represent each non-identity element not contained in D exactly ? times and each non-identity element contained in D exactly ?+β times. Such a set is closely related to association schemes of PBIB designs with two associate classes.

Partial Difference Triples

AbstractIt is known that a strongly regular semi-Cayley graph (with respect to a group G) corresponds to a triple of subsets (C, D, D′) of G. Such a triple (C, D, D′) is called a partial difference triple. First, we study the case when D ∪ D′ is contained in a proper normal subgroup of G. We basically determine all possible partial difference triples in this case. In fact, when

Some New Results on Circulant Weighing Matrices

|G| \ne 8

Strongly regular Cayley graphs with l−m=−1

nor 25, all partial difference triples come from a certain family of partial difference triples. Second, we investigate partial difference triples over cyclic group. We find a few nontrivial examples of strongly regular semi-Cayley graphs when |G| is even. This gives a negative answer to a problem raised by de Resmini and Jungnickel. Furthermore, we determine all possible parameters when G is cyclic. Last, as an application of the theory of partial difference triples, we prove some results concerned with strongly regular Cayley graphs.

On Circulant Complex Hadamard Matrices

Abelian relative difference sets of parameters (m, n, k, λ)=(pa, p, pa, pa−1)are studied in this paper. In particular, we show that for an abelian groupG of orderp2c+1and a subgroupN ofG of orderp, a (p2c, p, p2c, p2c−1)-relative difference set exists inG relative toN if and only if exp (G)≤pc+1.Furthermore, we have some structural results on (p2cp, p2c, p2c−1)-relative difference sets in abelian groups of exponentpc+1. We also show that for an abelian groupG of order 22c+2 and a subgroupN ofG of order 2, a (22c+1, 2, 22c+1, 22c)-relative difference set exists inG relative toN if and only if exp(G)≤2c+2 andN is contained in a cyclic subgroup ofG of order 4. New constructions of (p2c+1, p, p2c+1, p2c)-relative difference sets, wherep is an odd prime, are given. However, we cannot find the necessary and sufficient condition for this case.

Nonexistence of abelian difference sets: Lander’s conjecture for prime power orders

We obtain a few structural theorems for circulant weighing matrices whose weight is the square of a prime number. Our results provide new schemes to search for these objects. We also establish the existence status of several previously open cases of circulant weighing matrices. More specifically we show their nonexistence for the parameter pairs (n, k) (here n is the order of the matrix and k its weight) = (147, 49), (125, 25), (200, 25), (55, 25), (95, 25), (133, 49), (195, 25), (11 w, 121) for w < 62.

Constructions of Relative Difference Sets with Classical Parameters and Circulant Weighing Matrices

We obtain some results that are useful to the study of abelian difference sets and relative difference sets in cases where the self-conjugacy assumption does not hold. As applications we investigate McFarland difference sets, which have parameters of the form v=qd+1( qd+ qd-1 +...+ q+2) ,k=qd( qd+qd-1+...+q+1) , λ = qd ( q(d-1)+q(d-2)+...+q+1), where q is a prime power andd a positive integer. Using our results, we characterize those abelian groups that admit a McFarland difference set of order k-λ = 81. We show that the Sylow 3-subgroup of the underlying abelian group must be elementary abelian. Our results fill two missing entries in Kopilovichs table with answer “no”.