If $p^a \vert \vert n$ where $n >4$ is the order of a Circulant Hadamard matrix, then the order of $p$ modulo $n/p^a$ is odd

TL;DR

The paper proves a number-theoretic property of the order of primes related to circulant Hadamard matrices, leading to nonexistence results for such matrices and Barker sequences for infinitely many lengths.

Contribution

It provides a shorter, more general proof of a key property involving prime orders in circulant Hadamard matrices, extending previous results to include the prime 2.

Findings

Proved the odd order of prime modulo n/p^a for circulant Hadamard matrices.

Established nonexistence of circulant Hadamard matrices for infinitely many n.

Extended previous results to include the prime 2 using Brock's theorem.

Abstract

We proved recently (see \cite{lhgarasu}) the result on the title for odd prime divisors of such an $n.$ The result implies for many $n's$, more precisely, for an infinity of $n$'s with an arbitrary fixed number of prime divisors, the inexistence of circulant Hadamard matrices, and the inexistence of Barker sequences of length $n >13$. The proof used a result of Arasu. It turns out that there is another, shorter proof, of the more general result that includes the prime $p=2.$ This new proof is based on a result of Brock (see \cite[Theorem 3.1]{brock}), and besides that, requires just the definition of the Fourier transform. I noticed Brock's result in a preprint (see \cite{winterhofetal}) of Winterhof et al. where it is used to study the inexistence of related Butson-Hadamard matrices.