## Donald Preece Memorial Day

On Thursday we held a memorial day for Donald Preece at Queen Mary.

You can find a web page with the program and information here. This page will remain as a record for the time being. If you have anything interesting related to Donald (photos, videos, music, or just papers), please send it (or a link) to me and I will post it there.

I learned much about Donald that I hadn’t known before, even though I have read the obituary and bibliography carefully. But I don’t just want to tell the stories here. I will say a bit about some of the designs that interested Donald.

Let v and k be positive integers with v > k. We also require that there exist symmetric BIBDs with parameters (v,k,λ): a structure with v points and v blocks, any block incident with k points and any point with k blocks, and any two points incident with λ blocks and any two blocks with λ points.

The first kind of structure, which Donald referred to as a hyper-Graeco-Latin Youden “square”, is a k × v rectangle in which each entry is an m-tuple taken from an alphabet of size v, such that, for each position i in the m-tuple, the symbols in position i in the entries of the columns are the blocks of a symmetric BIBD. If m > 1, there is also a “linking” condition which is a bit complicated to state. Indeed, part of the trouble that Donald and I had with that first paper was that Donald’s version of the linking condition (coming from the fact that statistically certain estimators are required to have the same variance) is formally weaker than my version (which arose naturally in studying doubly transitive groups).

In my paper with Donald in 1976, we constructed an example with m = 3, k = 6, and v = 16. Indeed, we gave three mutually orthogonal 16×16 Latin squares such that the first six rows give an example (as also do the last ten rows, with k = 10).

The right way to think about this is that we have a set of vk “cells” or “plots”, with one partition into k sets of v (the rows) and m+1 partitions into v sets of k (the columns, and the symbols in each position in the m-tuples) such that the first partition is orthogonal to any of the others, while any two of the others define a symmetric BIBD and any three of them satisfy the linking condition. The other difficulty we had is that I didn’t understand the relevance of the partition with k parts, which played no role in my work on permutation groups. Indeed, it was more than twenty years later that I was able to understand the point of this partition, and construct an infinite family of examples.

But the other generalisation, one very close to Donald’s heart, takes more than one partition into k sets of v. This time we typically have m = 1 (so that there are only two of the v-part partitions), and v congruent to +1 or −1 (mod k), and we require that given parts of two of the k-part partitions, they intersect in either a or a+1 “plots”, for some a.

If there are two of the k-part partitions, this is what Donald called a double Youden rectangle. His most famous example has v = 13, k = 4, and can be realised with a pack of playing cards. This design was found by Donald in the early 1980s, and he was so pleased with it that he stuck the cards onto a cardboard backing and had them mounted:

To repeat explicitly the conditions,

• Each value A, 2, 3, …, 10, J, Q, K appears once in each row.
• Each suit appears once in each column.
• Each pair of the values A, 2, 3, …, 10, J, Q, K occurs together in just one column.
• Each row has 3 cards of each of 3 suits, and 4 cards of the other suit.

One of his last breakthroughs was a construction with more than two of the v-part partitions (so “triple, …, Youden rectangles”). This work is currently being written up by J. P. Morgan and should appear soon.

I do not know whether or not it is possible to have strictly more than two partitions of both types. No examples are known …