The cardinality of sets of k-independent vectors over finite fields

S. B. Damelin; G. Michalski; Gary L. Mullen

doi:10.1007/s00605-006-0440-6

The cardinality of sets of k-independent vectors over finite fields

S. B. Damelin, G. Michalski, Gary L. Mullen

Mathematical Sciences

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

7 Scopus citations

Abstract

A set of vectors is k-independent if all its subsets with no more than k elements are linearly independent. We obtain a result concerning the maximal possible cardinality Ind _q (n, k) of a k-independent set of vectors in the n-dimensional vector space F _q ⁿ over the finite field F _q of order q. Namely, we give a necessary and sufficient condition for Ind _q (n, k) = n + 1. We conclude with some pertinent remarks re applications of our results to codes, graphs and hypercubes.

Original language	English
Pages (from-to)	289-295
Number of pages	7
Journal	Monatshefte fur Mathematik
Volume	150
Issue number	4
DOIs	https://doi.org/10.1007/s00605-006-0440-6
State	Published - Apr 2007

Keywords

Bounds
Combinatorial design
Cycles
Finite fields
Girth
Graphs
Hypercubes
Linear codes
Linear independence

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10.1007/s00605-006-0440-6

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AU - Michalski, G.

AU - Mullen, Gary L.

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N2 - A set of vectors is k-independent if all its subsets with no more than k elements are linearly independent. We obtain a result concerning the maximal possible cardinality Ind q (n, k) of a k-independent set of vectors in the n-dimensional vector space F q n over the finite field F q of order q. Namely, we give a necessary and sufficient condition for Ind q (n, k) = n + 1. We conclude with some pertinent remarks re applications of our results to codes, graphs and hypercubes.

AB - A set of vectors is k-independent if all its subsets with no more than k elements are linearly independent. We obtain a result concerning the maximal possible cardinality Ind q (n, k) of a k-independent set of vectors in the n-dimensional vector space F q n over the finite field F q of order q. Namely, we give a necessary and sufficient condition for Ind q (n, k) = n + 1. We conclude with some pertinent remarks re applications of our results to codes, graphs and hypercubes.

KW - Bounds

KW - Combinatorial design

KW - Cycles

KW - Finite fields

KW - Girth

KW - Graphs

KW - Hypercubes

KW - Linear codes

KW - Linear independence

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SP - 289

EP - 295

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

IS - 4

ER -

The cardinality of sets of k-independent vectors over finite fields

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Keywords

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