# On computing all harmonic frames of $n$ vectors in $\C^d$

## Shayne Waldron and Nick Hay

## Abstract:

There are a finite number of
inequivalent isometric frames (equal--norm tight frames)
of $n$ vectors for $\Cd$ which are generated from a single vector
by applying an abelian group $G$ of symmetries.
Each of these so-called {\it harmonic frames}
can be obtained by taking $d$ rows of the character table of $G$;
often in many different ways,
which may even include using different %nonisomorphic
abelian groups.
Using an
algorithm implemented in the algebra package Magma,
we determine which of these are equivalent.
The resulting list of all harmonic frames for various choices of $n$ and $d$
is freely available,
and it includes many properties of the frames such as:
a simple description, which abelian groups generate it,
identification of the full group of symmetries,
the minimum, average and maximum distance between vectors in the frame,
and whether it is real or complex, lifted or unlifted.
Additional attributes aimed at specific applications include:
a measure of the cross correlation (Grassmannian frames),
the number of erasures (robustness to erasures),
and the diversity product of the full group of its symmetries
(multiple--antenna code design).
Some outstanding frames are identified and discussed, and
a number of questions are answered by considering the examples
on the list.

**Keywords:**
isometric frame,
equal--norm tight frame,
harmonic frame,
equiangluar frame,
Grassmannian frame

**Math Review Classification:**
Primary 42C15;
Secondary 20C15, 52B15

**Length:** 14 pages

**Last Updated:** 16 March 2006

## Availability: