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I am interested in finding out what is known about the following generalization of balanced incomplete block designs (BIBDs):

"What is the maximum size of a collection $B$ of $v$-dimensional unit real vectors with the following property: there exists a constant $\lambda$ such that $\forall x,y\in B$: $x\ne y \implies x\cdot y = \lambda$?"

Such a collection can be derived from a BIBD, but I wonder if larger collections exist?

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The Gram matrix of your vectors is equal to $(1-\lambda)I +\lambda J$ (where $J$ is the all-ones matrix). If $\lambda\ne -1/(v-1)$, this matrix is invertible, whence your set of vectors is linearly independent and there are at most $v$ of them. If $\lambda = -1/(v-1)$, then the Gram matrix has rank $v-1$ and your vectors are the $v+1$ vertices of a regular simplex.

The analogy to design theory is not compelling.

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  • $\begingroup$ Thanks! I suspected that nothing magical would happen here, but could not find a proof. $\endgroup$
    – Rasmus Pagh
    Commented Dec 21, 2015 at 13:33

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