The invariant factors of some cyclic difference sets

Recent progress in algebraic design theory

“…A more general decomposition result for group ring elements is proved by Leung and Schmidt [81]. Applications of this decomposition result include the nonexistence of circulant Hadamard matrices of order v with 4 < v < 548, 964, 900 and the nonexistence of Barker sequences of length with 13 < < 10 22 . For more details, we refer the reader to [81].…”

“…The following lemma is very useful for determining the SNF of (v, k, ) difference sets with gcd(v, k − ) = 1. [22]). Let G be an abelian group of order v, let p be a prime not dividing v, and let P be a prime ideal in Z[ v ] lying above p, where v is a complex primitive vth root of unity.…”

“…Using Lemma 9.3, Fourier transforms, and Stickelberger's congruence on Gauss sums, we [22] computed the number of 3's in the SNF of the Lin and HKM difference sets. …”

“…The work in [22] motivated us to ask whether it is true that two symmetric designs with the same parameters and having the same SNF are necessarily isomorphic. The answer to this question is negative.…”

The invariant factors of the incidence matrices of points and subspaces in 𝑃𝐺(𝑛,𝑞) and 𝐴𝐺(𝑛,𝑞)

“…Unfortunately, nonisomorphic designs sometimes have the same p-rank. In such a situation, one can try to prove nonisomorphism of designs by comparing the Smith normal forms of the incidence matrices [8]. Therefore it is of interest to find Smith normal forms of incidence matrices of designs.…”