2005 Hall Medal citation
Bulletin of the Institute of Combinatorics and its Applications,
May 2006, pp. 12–14
We have known Jonathan Jedwab for almost fifteen years, meeting him
regularly at many conferences. His talks have been consistently excellent;
indeed, it was always a pleasure to attend his lectures which can only be
described as highly polished, combining depth with clarity and elegance in
an extremely successful manner. Therefore, as a coorganizer the first
nominator invited him to give plenary talks at three conferences, namely
two Oberwolfach meetings (Codes and Designs in 1998 and Finite Geometries
in 2001) and the 2001 Capri conference on Discrete Mathematics and
its Industrial Applications. We have also followed his publications
closely, since we share many research interests.
Dr. Jedwab is no doubt one of the outstanding figures in Design Theory
today, where he has continuously considered really hard problems. Sadly,
as in Graph Theory, it is quite possible to produce lots of routine
papers in Design Theory, in particular in the purely combinatorial parts.
In contrast, Jonathan Jedwab's research always concerned topics like
difference sets which form a central and truly difficult part of Discrete
Mathematics, requiring familiarity with techniques from Geometry,
Combinatorial Theory, Group Theory, Representation Theory and Algebraic
Number Theory; and it is, of course, also closely related to some
important applied areas, in particular to Coding Theory and Signal
Processing. Indeed, his joint paper with Jim Davis in JCTA 80 (1997)
“A unifying construction for difference sets” can hardly be
overestimated in its importance for Algebraic Design Theory. Let us be a
bit more specific here. This paper is one of the most important ones in
the general area of difference sets ever. To mention just the most
It contains the first new parameter family of difference sets found in
almost 20 years (after Spence's 1977 paper).
It gives a unifying theory which produces a difference set with
gcd(v, n) > 1 in every abelian group which is known to contain
such a difference set. Not only was this true at the time of writing the
paper, but it even holds for the new families of difference sets which
were subsequently discovered by Chen.
It characterizes a certain class of groups containing McFarland
difference sets; such characterizations are what is really desired, but
they are extremely rare up to now and very hard to come by.
It allows a transparent treatment of the family of Hadamard difference
sets, whereas the best previous description (in terms of binary arrays
and binary supplementary quadruples, also due to Jedwab, based on his
Ph.D. thesis) was still rather cumbersome. In particular, the celebrated
characterization theorem for those abelian 2-groups which contain a
difference set becomes an almost trivial corollary; this problem had
taken decades to be settled.
It gives a unified way of constructing semiregular difference sets in
virtually all the known cases, and it provides many new examples.
When the difference set community (and it should be noted that this includes
— via the correspondence to sequences and arrays with good
correlation properties — also researchers from electrical
engineering) first learned about the paper, it created universal
excitement. E.g., the first nominator immediately decided to rewrite the
new difference sets chapter for the second edition of his “Design
Theory” (jointly with T. Beth and H.
Lenz, this appeared in the “Encyclopaedia of
Mathematics” series of Cambridge University Press in August 1999)
to include the new Davis-Jedwab theory of building sets. The paper has
become an instant classic and has already had enormous influence. Chen's
new constructions have to be seen in this framework, and Ionin's
important construction method for symmetric designs (which has yielded
many new infinite families) heavily relies on the Davis-Jedwab theory of
building sets. Similarly, there have been quite a few follow-up papers
concerned with semi-regular relative difference sets.
We would also like to emphasize that the unification paper is not an
isolated achievement in Jedwab's career, but is a particular highlight
in continuous work of exceptional quality. Let us mention just five more
The paper “Generalised perfect arrays and Menon difference
sets” (Designs, Codes and Cryptography 2 (1992), 19–68) gave
the first convincing approach to the theory of Menon difference sets,
and the binary supplementary quadruples used there are a precursor of
the theory developed in the unification paper.
The paper “Peak-to-mean power control in OFDM, Golay complementary
sequences, and Reed-Muller codes” (IEEE Trans. Inf. Theory 45
(1999), 2397–2417) gives a striking connection between seemingly
unrelated objects which is not only of great theoretical interest to
Coding and Design Theory but also solves an important real-world
Another outstanding example of Jedwab's way of coming up with interesting
mathematical results which solve an important real-world application was
his design for the IEEE 802.12 transmission code, about which he gave a
brilliant talk at the Oberwolfach meeting on “Finite
Geometries” in December 2001. For commercial reasons, this could
not be published for a long time, and is now finally submitted to IEEE
Trans. Comm. under the title “The design of the IEEE 802.12 coding
“Binary sequences with merit factor greater than 6.34” (IEEE
Trans. Inform. Th. 50 (2004), 3234–3249) contains sequences for
lengths up to several million with a merit factor larger than 6 (often
conjectured to be impossible).
Recently, in a paper submitted to the Proc. AMS “Proof of the
Barker array conjecture” a long standing open important problem
was solved in a surprisingly simple and elegant manner.
In our opinion Dr. Jedwab (who is now, after a long period as a senior
scientist at Hewlett Packard Labs in Bristol an Associate Professor of
Mathematics at Simon Fraser University) is one of the few mathematicians
whose work combines theoretical research of the highest quality and
originality with important real world applications. It should also be
stressed that creativity and originality alone do not suffice to produce
a coherent new theory as that given in the unification paper; such a
result also requires considerable stamina and perseverance to make the
vision become reality.
As this brief description of his work shows, Dr. Jedwab
not only has the potential to become an international leader in the
broad and important area of Discrete Mathematics and its applications in
Computer Science and Engineering, but is already well on his way to this
goal. His research shows a consistent quality and depth and has already
had a remarkable impact and will no doubt continue to do so. He amply
deserves being awarded a 2005 Hall medal.