Geodesic
The (non-)existence of perfect codes in Lucas cubes
Ars Mathematica Contemporanea, Tome 22 (2022) no. 3, article no. 10, 7 p.

Voir la notice de l'article provenant de la source Ars Mathematica Contemporanea website

The Fibonacci cube of dimension n, denoted as Γn, is the subgraph of the n-cube Qn induced by vertices with no consecutive 1’s. Ashrafi and his co-authors proved the non-existence of perfect codes in Γn for n ≥ 4. As an open problem the authors suggest to consider the existence of perfect codes in generalizations of Fibonacci cubes. The most direct generalization is the family Γn(1s) of subgraphs induced by strings without 1s as a substring where s ≥ 2 is a given integer. In a precedent work we proved the existence of a perfect code in Γn(1s) for n = 2p − 1 and s ≥ 3.2p − 2 for any integer p ≥ 2. The Lucas cube Λn is obtained from Γn by removing vertices that start and end with 1. Very often the same problems are studied on Fibonacci cubes and Lucas cube. In this note we prove the non-existence of perfect codes in Λn for n ≥ 4 and prove the existence of perfect codes in some generalized Lucas cube Λn(1s).
DOI : 10.26493/1855-3974.2308.de6
Keywords: Error correcting codes, perfect code, Fibonacci cube 
@article{10_26493_1855_3974_2308_de6,
     author = {Michel Mollard},
     title = {The (non-)existence of perfect codes in {Lucas} cubes},
     journal = {Ars Mathematica Contemporanea},
     eid = {10},
     publisher = {mathdoc},
     volume = {22},
     number = {3},
     year = {2022},
     doi = {10.26493/1855-3974.2308.de6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2308.de6/}
}
TY  - JOUR
AU  - Michel Mollard
TI  - The (non-)existence of perfect codes in Lucas cubes
JO  - Ars Mathematica Contemporanea
PY  - 2022
VL  - 22
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2308.de6/
DO  - 10.26493/1855-3974.2308.de6
LA  - en
ID  - 10_26493_1855_3974_2308_de6
ER  - 
%0 Journal Article
%A Michel Mollard
%T The (non-)existence of perfect codes in Lucas cubes
%J Ars Mathematica Contemporanea
%D 2022
%V 22
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2308.de6/
%R 10.26493/1855-3974.2308.de6
%G en
%F 10_26493_1855_3974_2308_de6
Michel Mollard. The (non-)existence of perfect codes in Lucas cubes. Ars Mathematica Contemporanea, Tome 22 (2022) no. 3, article  no. 10, 7 p. doi : 10.26493/1855-3974.2308.de6. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2308.de6/

Cité par Sources :