Comments on: Multiply transitive permutation sets

By: Vineet George

Vineet George — Sun, 22 Apr 2012 15:34:56 +0000

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By: Peter Cameron

Peter Cameron — Tue, 17 Apr 2012 07:24:48 +0000

The reason I think Delsarte's methods (and probably also semilinear programming) won't help is that the answer you want depends on the arithmetic structure of n, whereas the methods seem to pay no attention to this. As you know, a sharply 2-transitive set exists if n is a prime power but not if n is congruent to 2 or 3 (mod 4) and not the sum of two squares. The algebraic methods tend to give bounds uniform in n. I wonder if your new methods give (or can be adjusted to give) upper bounds for uniformly k-transitive sets. Maybe you can beat Bannai's method.

By: Bill Fahle

Bill Fahle — Mon, 16 Apr 2012 20:13:34 +0000

Agreed in the general case. I wonder, though, if we can prove lower bounds in specific cases where a sharp 2-transitive set is known not to exist, such as when n=6. Even finding an exact value for this one small case is proving impossible to compute via brute force, because of the C(720, ~30) search factor. The general upper bound is much more interesting, however, as we have found methods for building (n+1,k) sets from an (n,k) set and n copies of a (n,k-1) set, giving a fairly good upper bound, and we’re finding new theorems to improve this.

By: Peter Cameron

Peter Cameron — Sat, 14 Apr 2012 17:20:17 +0000

I think that the best reference to Delsarte's approach is his own exposition of it in his thesis, which was published more-or-less complete in a volume of Philips Research Reports Supplements in the early 1970s. You are right that the most important new idea to come along since Delsarte's thesis is semidefinite programming. I don't know for sure but I imagine that it can help with the lower bounds as well as the upper bounds. But as I said, I don't think that any general technique will, for example, improve on the lower bound n(n−1) for a 2-transitive permutation set, since this would be a painless way of showing that projective planes don't exist.

By: Dima

Dima — Sat, 14 Apr 2012 17:11:24 +0000

I realise that I don’t know anything about Delsarte’s approach to lower bounds. Could you give a reference?

By: Dima

Dima — Sat, 14 Apr 2012 17:07:34 +0000

as well, semidefinite programming based methods could in principle be applied here to certain coherent configurations (of which the assoc. schemes of conjugacy classes are sub-algebras). This has been carried out by Lex Schrijver in 2005 to improve bounds on “usual” binary codes, but it’s certainly applicable to permutation codes too.

By: Peter Cameron

Peter Cameron — Sat, 14 Apr 2012 17:07:28 +0000

Delsarte’s results give upper bounds for “codes” (where you specify the relations which are allowed to occur) and lower bounds on “designs” (where you specify the “strength” in a rather technical sense). I know Tarnanen’s paper; my point was that I don’t know any papers exploiting the lower bounds for permutation sets, or even interpreting what “strength” means for these).

By: Dima

Dima — Sat, 14 Apr 2012 16:24:56 +0000

Delsarte bounds on assoc. schemes of conjugacy classes of $S_n$ were applied by Tarnanen (Eur. J. Comb. 20(1999)), and he obtained in this way nontrivial bounds on permutation codes.
Lately coding theorists work on this more and more, I am told.

By: Bill Fahle

Bill Fahle — Mon, 09 Apr 2012 15:21:42 +0000

Incidentally, thanks for your reply and this helpful information. Here is a paper which describes an application of (n,k)-transet cardinality to fault tolerance in the permutation star graph. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4755773
At the time of this paper, the authors were unaware of the similarly-defined (equivalent) sharply k-transitive permutation sets.

By: Bill Fahle

Bill Fahle — Mon, 09 Apr 2012 12:44:11 +0000

I admit I was being a little coy. While the concept of uniformly transitive permutation sets didn't occur to me, the title of your post is the title of my freshly minted dissertation. Your construction is new to me also. We considered the minimal cardinality of any t-transitive set, defined as you have it. I called them (n, k)-transets, where k is your t. It is one of those things that computer scientists look at rather than mathematicians, because as you hinted, the sets are not always tidy, and are therefore difficult to write proofs about. However we have found a number of theorems giving constructions for non-trivial sets for all n and t (thus giving upper bounds), and also found lower bounds greater than 1+(n!/(n-k)!) for k >= 3 when there is nonexistence on sharply 2-transitive sets (such as through Bruck-Ryser, or Quistorff's result for 4<=n-k<=k, or Lam et al. for n=10). I'm writing up a paper for submission to ESA 2012 in Ljubjana, Slovenia. My current quest is to prove exact bounds for x=min |y| | y is a (6,2)-transet, which as of now we know 33<=x<=37.