Cyclotomic integers and finite geometry

Abstract:

We obtain an upper bound for the absolute value of cyclotomic integers which has strong implications on several combinatorial structures including (relative) difference sets, quasiregular projective planes, planar functions, and group invariant weighing matrices. Our results are of broader applicability than all previously known nonexistence theorems for these combinatorial objects. We will show that the exponent of an abelian group $G$ containing a $(v,k,\lambda ,n)$-difference set cannot exceed $\left (\frac {2^{s-1}F(v,n)}{n}\right )^{1/2}v$ where $s$ is the number of odd prime divisors of $v$ and $F(v,n)$ is a number-theoretic parameter whose order of magnitude usually is the squarefree part of $v$. One of the consequences is that for any finite set $P$ of primes there is a constant $C$ such that $\exp (G)\le C|G|^{1/2}$ for any abelian group $G$ containing a Hadamard difference set whose order is a product of powers of primes in $P$. Furthermore, we are able to verify Ryser’s conjecture for most parameter series of known difference sets. This includes a striking progress towards the circulant Hadamard matrix conjecture. A computer search shows that there is no Barker sequence of length $l$ with $13<l\le 4\cdot 10^{12}$. Finally, we obtain new necessary conditions for the existence of quasiregular projective planes and group invariant weighing matrices including asymptotic exponent bounds for cases which previously had been completely intractable.