What is the number of families of $4t+1$ subsets of order $2t$ of the set $\{1,2, \ldots ,p\}$, where $p$ is a prime number which is equal to $4t+1$ and the order of intersection of each pair of the subsets is $t$ or $t-1$ and each subset has $t$-intersection with $2t$ subsets and $t-1$-intersection with the other $2t$ subsets?



The number of specific structure
Yes, equivalently, the problem can be stated as follows: We are looking for the number of $p\times p$ 0-1-matrices $A$ with all row sums $2t$ such that $AA^T=tJ+tI-B$, where $B$ a 0-1-matrix with also all row sums equal to $2t$. (Here, $J$ denotes the all-1-matrix).


It is not hard to find *circulant* matrices that fulfill the conditions for each $p=4t+1$ (why only for primes?), and each of them gives rise to $(p-1)!/2$ set families. But different matrices can yield the same families, e.g. for $t=1$, we can have  

$A=circ(11000)$ with $AA^T=circ(21001)$ and  

$A=circ(10100)$ with $AA^T=circ(20110)$, each yielding the same 12 structures.


For $t=2$:  

$A=circ(111010000)$ with $AA^T=circ(4\ 2211\ 1122)$,  

$A=circ(110010100)$ with $AA^T=circ(4\ 1122\ 2211)$,  

$A=circ(110110000)$ with $AA^T=circ(4\ 2121\ 1212)$,  

$A=circ(110101000)$ with $AA^T=circ(4\ 1212\ 2121)$,  

$A=circ(111001000)$ with $AA^T=circ(4\ 2112\ 2112)$,  

$A=circ(110100010)$ with $AA^T=circ(4\ 1221\ 1221)$. 


$t=3$:  

$A=circ(1110110100000)$ with $AA^T=circ(6\ 333222\ 222333)$,  

$A=circ(1110101100000)$ with $AA^T=circ(6\ 332223\ 322233)$,  

$A=circ(1110110010000)$ with $AA^T=circ(6\ 323322\ 223323)$,  

$A=circ(1110011010000)$ with $AA^T=circ(6\ 322233\ 332223)$,  

$A=circ(1110100101000)$ with $AA^T=circ(6\ 232233\ 332232)$,  

and probably many others (looking at the patterns of $AA^T$). 


In fact, looking at the $t=2$ patterns that seem "complete", **I'd conjecture that there are $\binom{2t}{t}$ essentially different such circulant matrices**.  

(There may be also non-circulant solutions, so this would not yet solve the problem, but be a nice start.)


