Bounds on the maximum number of vectors with given scalar products

Deza, M.; Frankl, P.

doi:10.1090/S0002-9939-1985-0801348-0

Bounds on the maximum number of vectors with given scalar products
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by M. Deza and P. Frankl PDF

Proc. Amer. Math. Soc. 95 (1985), 323-329 Request permission

Abstract:

Suppose $M$, $L$ are sets of real numbers, $V = \{ {\upsilon _1}, \ldots ,{\upsilon _m}\}$ is a collection of vectors in ${R^n}$, having $k$ nonzero coordinates all from $M$ and satisfying $({\upsilon _i},{\upsilon _j}) \in L$ for $i \ne j$. Theorem 1.1 establishes a polynomial upper bound for $\left | V \right |$, generalizing previous results for subsets of a set and $(0, \pm 1)$-vectors. Theorem 1.4 asserts that if $\left | L \right | = s$ then $\left | V \right | \leqslant \left ( {\begin {array}{*{20}{c}} {n + s} \\ s \\ \end {array} } \right )$. For $M = \{ - 1,1\}$, $L = [ - (k - 1),k - 1]$, Theorem 1.5 gives $\left | V \right | \leqslant {2^{k - 1}}\left ( {\begin {array}{*{20}{c}} n \\ {k - 1} \\ \end {array} } \right )/k$, where equality holds if and only if $V$ is a "signed" $(n,k,k - 1)$ Steiner-system.

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