2005 Hall Medal citation

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 striking results:

• 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 examples:

• 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 problem.
• 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 scheme”.
• “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.