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.

A survey of partial difference sets

All Two-Weight Irreducible Cyclic Codes?

The most powerful method for the study of finite geometries with regular or quasiregular automorphism groups G is to translate their definition into an equation over the integral group ring 2[G] and to investigate this equation by applying complex representations of G. For the definitions and the basic facts, see Section 2. If G is abelian, this approach boils down to proving or disproving the existence of elements of 2[G] with 0-1 coefficients whose character values are cyclotomic integers of certain prescribed absolute values. Up to now there have been two general methods to tackle this problem, namely, Hall's multiplier concept and Turyn's selfconjugacy approach. However, both methods need severe technical assumptions and thus are not applicable to many classes of problems. Despite many efforts over a period of more than 30 years no general method has been found to overcome these difficulties. In this paper, we present a new approach to the study of combinatorial structures via group ring equations which works without any restrictive assumptions. In order to understand the method of the present paper it will be instructive to briefly discuss the self-conjugacy concept first. Turyn [51] demonstrated that the character method for the study of group ring equations works very nicely under the so-called self-conjugacy condition. An integer n is called self-conjugate modulo m if all prime ideals above n in the mth cyclotomic field Q((m) are invariant under complex conjugation. Under this condition it is possible to find all cyclotomic integers in Q((m) of absolute value nt/2 for any positive integer t. It is the complete knowledge of the cyclotomic integers of prescribed absolute value which makes the character method work so well under the self-conjugacy condition. Since Turyn's fundamental work [51] there have been dozens of papers extending and refining his approach. However, all these results are restricted to the case of self-conjugacy, and that is a very severe restriction indeed. Namely, the "probability" that n is selfconjugate modulo m decreases exponentially fast in the number of distinct prime divisors of n and m; see Remark 2.2. This means that the self-conjugacy method fails in almost all cases. One may ask if it is possible to extend Turyn's method in order to get rid of the self-conjugacy assumption. It turns out that in general this is impossible at least with present day methods. The required complete knowledge of the cyclotomic integers of prescribed absolute value would yield an

/pdf/cyclotomic-integers-and-finite-geometry-1o84w2yzw2.pdf

Cyclotomic integers and finite geometry

We obtain several new number theoretic results which improve the field descent method. We use these results to rule out many of the known open cases of the circulant Hadamard matrix conjecture. In particular, the only known open case of the Barker sequence conjecture is settled.

https://www3.ntu.edu.sg/home/Bernhard/Publications/pub/chdm.pdf

R. L. McFarland

Papers

Sub-difference sets of Hadamard difference sets