I am interested in proving that the incidence matrix M for any symmetric balanced incomplete block design can be written in a manner where M equals M transpose. If so then the incidence matrix will also be symmetric.
This is not true in general. For example, there are precisely 4 projective planes of order 9, which are symmetric (v,k,lambda) = 2-(91,10,1)-designs. Self-duality corresponds to the property for incidence matrices of symmetricity. Two of these planes are self-dual: the desarguesian plane PG(2,9), and the Hughes plane,
but the other ones are the nearfield plane and the dual of this plane: so these two are not self-dual. Actually this is the smallest example for finite projective planes, since all the smaller planes (of prime-power orders) are desarguesian, isomorphic to PG(2,q) for GF(q) a finite field, and so are self-dual. "Symmetric for 2-designs means that the number of varieties equals the number of blocks (v=b), but this is not the same as self-duality, where the roles of varieties and blocks is interchanged.
You can find a paper by Wallis and Lin from some 20 years ago or so in which it is shown that there are Hadamard matrices of various orders which are not symmetric-equivalent. This is much stronger than saying that the related SBIBDs are not symmetric. So what you're looking for is not even close to being true.
What is a credible hypothesis -- but probably far too hard for anyone in our lifetime to prove -- is that whenever an SBIBD exist, there also exists one of the same parameters with a symmetric incidence matrix.
I have been using times two plus one to generate infinte families of what I call super-symmetric block designs. Can construct simple 0,1,3 =llamda,k,v design.Then using times two plus one can get 3,7,15 and then 7,15,31. The sequence 0,1,3,7,15,31,63,---
yields an infinite number of designs in which the incidence matrix M equals it's transpose. So M^2 can be used to verify that it is super-symmetric. Am working on 2,5,11,23,47,---- and so on. As long as 2,5,11 is supersymm then so are all others. Since the designs with odd starters are included in those with even starters only have to consider those with even staters from now on.
4,9,19,39,-----
6,13,27,55----
8,17,35,71,---
and so on forever.
would be nice to show (2n,4n+1,8n+3) exists in general. Then of course all Hadamards of order 4n would exist.It is pleasing to dream of these things wether or not they can shownn to exist.
Will send youa copy of paper once it is published on new designs implied from symmetric block designs.
BIBDs with the first list of parameters and a symmetric incidence matrix are well known. They come from the so-called Sylvester-Hadamard matrices given in Sylvester's 1867 paper. If you're using a "2n+1" construction then you're being too complicated. It's a very simple application of block doubling.
The second follows from the H(12) constructed by Hadamard in 1893 plus the same doubling construction. The others you list follow from known symmetric Hadamard matrices in the same way. These are known for many small orders. I'm not sure at which point we do not have a symmetric one, but the first currently unresolved order of Hadamard matrices is 668. So ... find one of these designs of order 667 and you will be famous. Good luck :-)
Rob: Will soon be publishing paper on existence of unknown symmetric block designs. Have included designs where v=667,715,891,1003,1131,1243,1387,1435,1675,1771,1915,1947,1963. From these Hadamards of order v+1 can then be formed. I am not saying that I can construct them but showing the good possibility of their existence is what I am doing.
"good probability", in what sense? Can we quantify this, or is this a heuristic argument for existence?
Rob: At present am attempting to build (216,16,2) . The existence of this design implies the existence of a much larger design which has been constructed by a different algorithm. I am attempting to reverse engineer this design into (216,16,2). Have chosen this one since it appears to be smallest and simplest from list of designs whose existence are still in question.Will let you know how I make out with this problem. This same concept will apply to (667,333,166) and the others as well.
A reasonable thing to try. However, you know that it can be very hard to reverse engineer such things in an ad hoc fashion. The H(428), for example, was found by a direct construction. For many years we had H(856), but nobody was successful in reverse engineering to get H(428) even though the most basic propagation method doubles the order. Intuitively, one would think, "to get H(428) just 'undouble' H(856)". However as we know the VAST majority of H(2n)s do not come from H(n) using any known construction, so until something new is learned, doing this on an ad hoc basis may be a fool's errand.
Of course if you know that ALL designs of the "known" type are derived in known ways from SOME design of the "unknown" type then this approach is quite likely to pay off.
Rob: Have just created another type of design from symmetric block designs. If Symmetric design exists then so does this one . Was hoping for a contradiction here with some of the designs whose existence remains in question, No such luck.
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