Solutions

The word “solutions” is much overused, even misused, now. When I see a van with “Cleaning solutions” on the side, I imagine it full of containers of ammonia or soapy water, while “Printing solutions” can only mean ink … But here I am talking about “solutions to the exercises”.

Designs, Graphs, Codes and their Links is the only book I wrote jointly. It came about when Jack van Lint and I were separately invited by Dan Hughes to lecture at Westfield College on things connected to designs (the audience, including Fred Piper and Marion Kimberley, were experts on design theory). I chose graphs, since this was soon after I had written a thesis on the Higman–Sims graph and related things. Jack knew about codes, which give rise to designs by taking the supports of codewords of given weight.

After the lectures, Dan suggested that we should publish them jointly as a book in the London Mathematical Society Lecture Note series. We put them together with minimal editing; I think the typing was done at Westfield.

The book went through several versions. The publishers, Cambridge University Press, preferred a new title to a new edition with the same title, so we went through several permutations of graphs, codes and designs. Finally in 1990 the Press proposed moving the book to the Student Texts series. This meant, as well as a major revision, adding a large number of exercises. We planned this at the Marshall Hall memorial conference in Burlington, Vermont, in 1990, singlemindedly missing the excursion (which included a trip to Ben and Jerry’s factory) to sit in a motel room all afternoon dividing up the workload and suggesting new topics and problems.

The book contains material which can’t easily be found elsewhere. But if there were to be a new edition, it would need another major rewrite. For example, we treat Kerdock codes and the associated binary geometry; but we were too early for the major paper of Hammons et al. on quaternary codes and the Gray map, followed by Calderbank et al. on the connections with real and complex line systems, from which there is a direct link to mutually unbiased bases in quantum computing.

At some point, somebody emailed me to ask whether there were solutions to the exercises. I sat down and wrote out solutions to the exercises in the first eight chapters (essentially, the ones I wrote), and at some later point I typed these up in LaTeX.

I had completely forgotten about this until, going through old files, I came upon them. I could make them public now, though they will need a careful check first.

I’m not rushing in because I know there are different views on solutions to exercises. Basically, if you are studying the book on your own, they are very helpful, though there is always a temptation to look at the solutions too soon, or to say “I know how to do that, I will just check the solution” rather than engaging fully with the problem. On the other hand, many instructors using a textbook like to assign problems to the class, and the last thing they want is to have the solutions available to the students.

I suspect that DGCL is used more in the first way than in the second. Any thoughts?