# Fixed Points of Boolean Networks, Guessing Graphs, and Coding Theory

Abstract : In this paper, we are interested in the number of fixed points of functions $f:A^n\to A^n$ over a finite alphabet $A$ defined on a given signed digraph $D$. We first use techniques from network coding to derive some lower bounds on the number of fixed points that only depends on $D$. We then discover relationships between the number of fixed points of $f$ and problems in coding theory, especially the design of codes for the asymmetric channel. Using these relationships, we derive upper and lower bounds on the number of fixed points, which significantly improve those given in the literature. We also unveil some interesting behaviour of the number of fixed points of functions with a given signed digraph when the alphabet varies. We finally prove that signed digraphs with more (disjoint) positive cycles actually do not necessarily have functions with more fixed points.
Type de document :
Article dans une revue
Siam Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2015, 〈10.1137/140988358〉
Domaine :
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https://hal.archives-ouvertes.fr/hal-01298045
Contributeur : Adrien Richard <>
Soumis le : mardi 5 avril 2016 - 13:33:10
Dernière modification le : mercredi 31 janvier 2018 - 10:24:05
Document(s) archivé(s) le : mercredi 6 juillet 2016 - 13:52:32

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1409.6144v1.pdf
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Maximilien Gadouleau, Adrien Richard, Søren Riis. Fixed Points of Boolean Networks, Guessing Graphs, and Coding Theory. Siam Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2015, 〈10.1137/140988358〉. 〈hal-01298045〉

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