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1. CJM 1999 (vol 51 pp. 326)
Association Schemes for Ordered Orthogonal Arrays and $(T,M,S)$-Nets In an earlier paper~\cite{stinmar}, we studied a generalized Rao bound
for ordered orthogonal arrays and $(T,M,S)$-nets. In this paper,
we extend this to a coding-theoretic approach to ordered orthogonal
arrays. Using a certain association
scheme, we prove a MacWilliams-type theorem for linear ordered orthogonal
arrays and linear ordered codes as well as a linear programming bound
for the general case. We include some tables which compare this
bound against two previously known bounds for ordered orthogonal arrays.
Finally we show that, for even strength, the LP bound is always at
least as strong as the generalized Rao bound.
Categories:05B15, 05E30, 65C99 |
2. CJM 1998 (vol 50 pp. 739)
Eigenpolytopes of distance regular graphs Let $X$ be a graph with vertex set $V$ and let $A$ be
its adjacency matrix. If $E$ is the matrix representing orthogonal
projection onto an eigenspace of $A$ with dimension $m$, then $E$ is
positive semi-definite. Hence it is the Gram matrix of a set of $|V|$
vectors in $\re^m$. We call the convex hull of a such a set of vectors
an eigenpolytope of $X$. The connection between the properties of this
polytope and the graph is strongest when $X$ is distance regular and,
in this case, it is most natural to consider the eigenpolytope
associated to the second largest eigenvalue of $A$. The main result
of this paper is the characterisation of those distance regular graphs
$X$ for which the $1$-skeleton of this eigenpolytope is isomorphic to
$X$.
Categories:05E30, 05C50 |