It is well known that for a number of structures, their existence is equivalent to the existence of a projective plane for a given order. Some of them depend on more than one parameters, which means that there can be several ways of measuring "how close" one can approach a (conjecturally non existing) projective plane for a given order $n$ with more than one prime divisor.

To illustrate this, take e.g. the biggest number $a(n)$ of mutually orthogonal latin squares (MOLS) of order $n$. For prime order, we have trivially $a(n)=n-1$, and moreover non isomorphic sets of $n-1$ MOLS correspond to non isomorphic projective planes of order $n$. On the other hand, it is well known that for $n=6$, there are not even two MOLS, i.e. $a(6)=1$, further that $2\le a(10)\le 6$. This survey is not very recent but gives a lot of lower bounds. The well-known fact that $a(12)\ge5$ means e.g. that for $n=12$, one can come "much closer" (in a certain sense) to a projective plane than for $n=6$.

A different way of measuring is invoked in this recent thread.

Which other structures allow to "measure" how close one can get?

(I would have thought that such a collection already exists in MO, but I couldn't find any.)


1 Answer 1


One equivalent structure which I like very much comes from the following old (and easy) theorem of Witt: There is a projective plane of order $n$ if and only if there is a sharply $2$-transitive set of permutations on a set of size $n$.

A permutation code is a subset $S$ of the symmetric group $S_n$, where the distance of two permutations $x$ and $y$ is $n$ minus the number of fixed point of $xy^{-1}$.

So if $S$ is sharply $2$-transitive, then $xy^{-1}$ has at most one fixed point for distinct elements $x,y\in S$, and $\lvert S\rvert=n(n-1)$. So the minimal distance of $S$ is $\ge n-1$.

On the other hand, it is easy to see that a permutation code $S\subseteq S_n$ of minimal distance $\ge n-1$ has at most $n(n-1)$ elements, and equality holds if snd only if $S$ is sharply $2$-transitive.

So the existence of finite projective planes is equivalent to the existence of certain permutation codes.


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