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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bounds on the maximum number of vectors with given scalar products


Authors: M. Deza and P. Frankl
Journal: Proc. Amer. Math. Soc. 95 (1985), 323-329
MSC: Primary 52A37; Secondary 05C35, 51K99
MathSciNet review: 801348
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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\vert V \right\vert$, generalizing previous results for subsets of a set and $ (0, \pm 1)$-vectors. Theorem 1.4 asserts that if $ \left\vert L \right\vert = s$ then $ \left\vert V \right\vert \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\vert V \right\vert \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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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0801348-0
PII: S 0002-9939(1985)0801348-0
Article copyright: © Copyright 1985 American Mathematical Society