# Algebraic Design Theory with Hadamard Matrices: Applications, Current Trends and Future Directions (14w2199)

Arriving in Banff, Alberta Friday, July 11 and departing Sunday July 13, 2014

## Organizers

(University of Manitoba)

(National University of Ireland, Galway (Ireland))

(University of Lethbridge)

## Objectives

At this critical juncture we propose bringing together some of the
key established players and emerging stars, to explore these new
directions, share with one another the state of the art, and take
stock of what remains to be done on classical problems.

Talks will focus primarily on (1) new methods of attack for old
problems; (2) new structures spawned out of the central field;
(3) new paradigms introduced in the recent books; (4) applications,
especially in quantum information theory; and (5) a vision for
setting the direction for future work. Each participant will have
the opportunity to speak. There will be problem-solving sessions,
and breakout sessions on questions under all the above headings.

Many of the proposed attendees have previously collaborated to
solve substantial problems in algebraic design theory. We envisage
the workshop as a platform event in the growth of the
international design theory community, via cross-fertilization of
ideas from seasoned mathematicians, early-stage researchers, and
doctoral students.

We now mention several particularly salient problems that would
feature at the workshop. It is certain that other problems, and
progress on them, will arise naturally through interaction between
participants.

Asymptotic existence results for Hadamard matrices are of the
following kind: for all odd positive integers $s$, there is a
Hadamard matrix of order $2^m s$ where $m\leq a_0+b_0\log_2 s$
($a_0$ and $b_0$ are non-negative constants independent of $s$).
Seberry showed that $(a_0, b_0) = (0, 2)$ is valid. Later work by
Craigen, Holzmann, and Kharaghani allows one to take $b_0=3/8$.
Current developments, including the use of sieve methods from
number theory by Smith, suggest that the exponent can be further
reduced. This would represent a significant advance on proving the

In their book, de Launey and Flannery defined the unifying notion
of pairwise combinatorial design $D$ for an orthogonality set
$\Lambda$ (of which Hadamard matrices are a special case). Such
designs $D$ are solutions of a Gram matrix equation $DD^* = b$
over some ambient ring $R$, where $b$ lies in a finite set
determined by $\Lambda$ and $R$. As this constraint is quadratic,
it will have an appropriate local-to-global principle (in analogy
with the Hasse-Minkowski theorem). Such a viewpoint leads, for
example, to the famous Bruck-Ryser-Chowla theorem. We seek to
formulate a suitable local-to-global theory for general
orthogonality sets.

The condition $DD^* = b$ translates into a group ring (or norm)
equation $a\overline{a}=k$, whose solutions $a \in \mathbb{Z}[\zeta_m]$ for a primitive $m$th root of unity $\zeta_m$
correspond to important associated objects such as (relative)
difference sets. In turn, the latter yield sequences with good
correlation properties. Schmidt's `descent method' describes
solutions of this equation in terms of the prime divisors of $m$.
It is also known how to solve the equation when $m$ is prime. But
the general problem of obtaining effective algorithms for solving
the group ring equation remains open.

A possible constructive approach to the Hadamard conjecture based
on $2$-cohomology of finite groups was initiated by de Launey and
Horadam in the early 1990s. There is still much scope for bringing
algebraic tools to bear on existence and classification problems
here. For example, a cocyclic Hadamard matrix indexed by a group
$G$ exists if and only if there is a non-trivial ring homomorphism
from the group ring $\mathbb{C}R_2(G)$ to the complex field
$\mathbb{C}$, where $R_2(G)$ is a finitely generated abelian group
whose relations come from the $2$-cocycle defining equation. The
kernels of such homomorphisms are amenable to further study.