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{\bf Thomas W. Cusick}
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{\bf Finite Vector Spaces and Certain Lattices}
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The Galois number $G_n(q)$ is defined to be the number of subspaces of the
$n$-dimensional vector space over the finite field $GF(q)$. When $q$ is
prime, we prove that
$G_n(q)$ is equal to the number $L_n(q)$ of $n$-dimensional mod $q$
lattices, which are defined to be lattices (that is, discrete additive
subgroups of n-space) contained in the integer lattice ${\bf Z}^n$ and
having the property that given any point $P$ in the lattice, all points of
${\bf Z}^n$ which are congruent to $P$ mod $q$ are also in the lattice.
For each $n$, we prove that $L_n(q)$ is a multiplicative function of $q$.
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