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 $q$.
2019 ◽
Vol 19
(05)
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pp. 2050086
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2009 ◽
Vol DMTCS Proceedings vol. AK,...
(Proceedings)
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2011 ◽
Vol 85
(1)
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pp. 19-25
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1985 ◽
Vol 98
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pp. 139-156
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2019 ◽
Vol 53
(1 (248))
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pp. 23-27
2017 ◽
Vol 16
(01)
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pp. 1750007
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