The existence of perfect codes in a family of generalized Fibonacci cubes

Michel Mollard (1)

1 IF - Institut Fourier

The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma_n$, is the subgraph of the $n$-cube $Q_n$ induced by vertices with no consecutive 1's. In an article of 2016 Ashrafi and his co-authors proved the non-existence of perfect codes in $\Gamma_n$ for $n\geq 4$. As an open problem the authors suggest to consider the existence of perfect codes in generalization of Fibonacci cubes. The most direct generalization is the family $\Gamma_n(1^s)$ of subgraphs induced by strings without $1^s$ as a substring where $s\geq 2$ is a given integer. We prove the existence of a perfect code in $\Gamma_n(1^s)$ for $n=2^p-1$ and $s \geq 3.2^{p-2}$ for any integer $p\geq 2$.

Mots clés: Fibonacci cube; Perfect code; Error correcting codes; Perfect codes in graphs

Domaines: Combinatoire [math.CO]

Michel Mollard. The existence of perfect codes in a family of generalized Fibonacci cubes. Information Processing Letters, 2018, 140, pp.1-3. ⟨10.1016/j.ipl.2018.07.010⟩. ⟨hal-01677671⟩