Finite vector spaces and certain lattices

Thomas W. Cusick

doi:10.37236/1355

Finite vector spaces and certain lattices

Thomas W. Cusick

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

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 Z ⁿ and having the property that given any point P in the lattice, all points of Zⁿ 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.

Original language	English
Pages (from-to)	18DUMMY
Journal	Electronic Journal of Combinatorics
Volume	5
Issue number	1
DOIs	https://doi.org/10.37236/1355
State	Published - 1998

Keywords

Galois numbers
Identities
Lattice
Multiplicative function
Vector space

Access to Document

10.37236/1355

Cite this

@article{8b2546db6dae4bc29ee38f3e358d7961,

title = "Finite vector spaces and certain lattices",

abstract = "The Galois number Gn(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 Gn(q) is equal to the number Ln(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 Z n and having the property that given any point P in the lattice, all points of Zn which are congruent to P mod q are also in the lattice. For each n, we prove that Ln(q) is a multiplicative function of q.",

keywords = "Galois numbers, Identities, Lattice, Multiplicative function, Vector space",

author = "Cusick, \{Thomas W.\}",

year = "1998",

doi = "10.37236/1355",

language = "English",

volume = "5",

pages = "18DUMMY",

journal = "Electronic Journal of Combinatorics",

issn = "1097-1440",

publisher = "Australian National University",

number = "1",

}

TY - JOUR

T1 - Finite vector spaces and certain lattices

AU - Cusick, Thomas W.

PY - 1998

Y1 - 1998

N2 - The Galois number Gn(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 Gn(q) is equal to the number Ln(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 Z n and having the property that given any point P in the lattice, all points of Zn which are congruent to P mod q are also in the lattice. For each n, we prove that Ln(q) is a multiplicative function of q.

AB - The Galois number Gn(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 Gn(q) is equal to the number Ln(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 Z n and having the property that given any point P in the lattice, all points of Zn which are congruent to P mod q are also in the lattice. For each n, we prove that Ln(q) is a multiplicative function of q.

KW - Galois numbers

KW - Identities

KW - Lattice

KW - Multiplicative function

KW - Vector space

UR - https://www.scopus.com/pages/publications/0346898887

U2 - 10.37236/1355

DO - 10.37236/1355

M3 - Article

AN - SCOPUS:0346898887

SN - 1097-1440

VL - 5

SP - 18DUMMY

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 1

ER -

Finite vector spaces and certain lattices

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this