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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Extending the $ t$-design concept

Authors: A. R. Calderbank and P. Delsarte
Journal: Trans. Amer. Math. Soc. 338 (1993), 941-952
MSC: Primary 05E30; Secondary 05B05, 33C45
MathSciNet review: 1134756
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Abstract: Let $ \mathfrak{B}$ be a family of $ k$-subsets of a $ v$-set $ V$, with $ 1 \leq k \leq v/2$. Given only the inner distribution of $ \mathfrak{B}$, i.e., the number of pairs of blocks that meet in $ j$ points (with $ j = 0,1, \ldots ,k$), we are able to completely describe the regularity with which $ \mathfrak{B}$ meets an arbitrary $ t$-subset of $ V$, for each order $ t$ (with $ 1 \leq t \leq v/2$). This description makes use of a linear transform based on a system of dual Hahn polynomials with parameters $ v$, $ k$, $ t$. The main regularity parameter is the dimension of a well-defined subspace of $ {\mathbb{R}^{t + 1}}$, called the $ t$-form space of $ \mathfrak{B}$. (This subspace coincides with $ {\mathbb{R}^{t + 1}}$ if and only if $ \mathfrak{B}$ is a $ t$-design.) We show that the $ t$-form space has the structure of an ideal, and we explain how to compute its canonical generator.

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Keywords: $ t$-design, $ t$-form space, $ t$-distribution matrix, dual Hahn polynomials
Article copyright: © Copyright 1993 American Mathematical Society

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