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Short covering codes arising from matchings in weighted graphs


Authors: Anderson N. Martinhão and Emerson L. Monte Carmelo
Journal: Math. Comp. 82 (2013), 605-616
MSC (2010): Primary 11B75, 05C70, 94B75; Secondary 05B40, 11T71, 94B25
DOI: https://doi.org/10.1090/S0025-5718-2012-02613-5
Published electronically: May 1, 2012
MathSciNet review: 2983038
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Abstract: The concept of embedded matching in a weighted graph is introduced, and the maximum cardinality of an embedded matching is computed. On the other hand, consider the following problem induced by a short covering. Given a prime power $ q$, the number $ c(q)$ denotes the minimum cardinality of a subset $ \mathcal {H}$ of $ \mathbb{F}_q^3$ which satisfies the following property: every element in this space differs in at most $ 1$ coordinate from a scalar multiple of a vector in $ \mathcal {H}$. As another goal, a connection between embedded matching and short covering code is established. Moreover, this link is applied to improve the upper bound on $ c(q)$ for every odd prime power $ q$.


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Additional Information

Anderson N. Martinhão
Affiliation: Departamento de Matemática, Universidade Estadual de Maringá, Brazil
Email: and{\textunderscore}nm@hotmail.com

Emerson L. Monte Carmelo
Affiliation: Departamento de Matemática, Universidade Estadual de Maringá, Brazil
Email: elmcarmelo@uem.br

DOI: https://doi.org/10.1090/S0025-5718-2012-02613-5
Keywords: Matching, weighted graph, square number, finite field, independent vectors, covering codes.
Received by editor(s): April 12, 2011
Received by editor(s) in revised form: August 24, 2011
Published electronically: May 1, 2012
Additional Notes: The first author was supported by Capes.
The second author is supported by Fundação Araucária and CNPq.
Dedicated: This work is dedicated to Professor Adilson Gonçalves
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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