Bilinear Forms on Finite Abelian Groups and GroupInvariant Butson Hadamard Matrices
Abstract
Let $K$ be a finite abelian group and let $\exp(K)$ denote the least common multiple of the orders of the elements of $K$. A $BH(K,h)$ matrix is a $K$invariant $K\times K$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=KI$, where $H^*$ denotes the complex conjugate transpose of $H$, and $I$ is the identity matrix of order $K$. Let $\nu_p(x)$ denote the $p$adic valuation of the integer $x$. Using bilinear forms on $K$, we show that a $BH(K,h)$ exists whenever (i) $\nu_p(h) \geq \lceil \nu_p(\exp(K))/2 \rceil$ for every prime divisor $p$ of $K$ and (ii) $\nu_2(h) \ge 2$ if $\nu_2(K)$ is odd and $K$ has a direct factor $\mathbb{Z}_2$. Employing the field descent method, we prove that these conditions are necessary for the existence of a $BH(K,h)$ matrix in the case where $K$ is cyclic of prime power order.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.07310
 Bibcode:
 2019arXiv190307310D
 Keywords:

 Mathematics  Combinatorics