Let $(v, k, \lambda)$ be a design $\mathcal{D}$. We say it's symmetric if $b=v$ where $b$ is the number of blocks in $\mathcal{D}$. A design also can be realized also by constructing the incidence matrix of dimensions $b \times v$.
I am puzzled by the claim that follows corollary 2.4 in "Combinatorial designs. Constructions and analysis", Douglas Stinson.
Corollary 2.4. Suppose M is the incidence matrix of a symmetric (v, k, λ)-BIBD. Then $M^T$ is also the incidence matrix of a (symmetric) (v, k, λ)-BIBD
Corollary 2.4 says that the dual of a symmetric BIBD is again a symmetric BIBD. We note that these two BIBDs need not be identical or even isomorphic
I found examples of $M \not = M^T$ but I could not find any example where $M \not \cong M^T$.
What I've tried:
- Search for small examples of symmetric design
- Construct the incidence matrix and its transpose
- Create a graph in which there is an edge between $(i,j)$ iff the element $i$ belongs to the block $j$ for $M$ and $M^T$ (Let's call them $G, G_T$
- Check if $G \cong G_T$ using sage
Based on the experimental results I'm inclined to believe $M \cong M^T$ but could not prove that as well.
Another update:
Morgan Rodgers suggested checking projective planes of order 9.
Since taking transpose turns every point into a line and every line into a point. According to Wikipedia there are non-daul projective planes i.e. turning every line into a point and vice versa does not result in an isomorphic projective plane.
According to wikipedia Hugh's planes, so the dual construction is not isomorphic to the original one.
One also, can use sagemath to construct such example using the command
sage.combinat.designs.block_design.HughesPlane(q2, check=True)
See here
Morgan Rodgers has already answered this in the comments, but there are additional remarks to be made that may be worthwhile.
First, there aren't all that many viable "small" parameter sets. We can ignore the trivial designs with $k=1$ or $k=v-1$. Since a design is self-dual if and only its complementary design is self-dual, we may further restrict our attention to $1<k<v/2$. Define the order of a symmetric BIBD by $n=k-\lambda$ (which is compatible with the notion of order for finite projective planes). All solutions to $(v-1)(k-n)=k(k-1)$ for a given $n$ that are not ruled out by the Bruck-Ryser-Chowla Theorem are listed below for the lowest values of $n$. The notations P, H, and M denote finite projective planes, Hadamard designs, and Menon designs. $$ \begin{array}{r|l} n & (v,k,\lambda)\\ \hline 1 & \text{trivial designs}\\ 2 & (7,3,1)\text{ (P,H)}\\ 3 & (13,4,1)\text{ (P)},\ (11,5,2)\text{ (H)}\\ 4 & (21,5,1)\text{ (P)},\ (16,6,2)\text{ (M)},\ (15,7,3)\text{ (H)}\\ 5 & (31,6,1)\text{ (P)},\ (19,9,4)\text{ (H)}\\ 6 & (25,9,3),\ (23,11,5)\text{ (H)}\\ 7 & (57,8,1)\text{ (P)},\ (37,9,2),\ (31,10,3),\ (27,13,6)\text{ (H)} \end{array} $$ All of the projective planes on the list above are unique up to isomorphism, and therefore self-dual. The same is true of the Hadamard design $(11,5,2)$. There are three $(16,6,2)$ Menon designs up to isomorphism. All three are self-dual. The three designs are, in fact, distinguished by having different $2$-ranks; since a design and its dual have the same $2$-rank, self-duality of these designs follows. (The $2$-rank is the rank of the incidence matrix, with arithmetic performed mod $2$.)
This brings us to the $(15,7,3)$ Hadamard designs. There are five of these, three of them self-dual. The other two are the design with incidence matrix below, and its dual, which is not isomorphic to it. $$ \left( \begin{array}{ccccccccccccccc} 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 \\ \end{array} \right) $$
The $(25,9,3)$ designs are instructive. It was established by Denniston that there are $78$ of these up to equivalence. Of these $22$ are self-dual; the other $56$ occur in $28$ dual pairs. For the Hadamard designs, there is an explosion in the number of designs, starting with parameters $(31,15,7)$. The overwhelming majority of these designs are not self-dual.