# 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

Robert Craigen (University of Manitoba)

Dane Flannery (National University of Ireland, Galway (Ireland))

Hadi Kharaghani (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

Hadamard conjecture.

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.

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

Hadamard conjecture.

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.