Perfect codes in generalized Fibonacci cubes

TL;DR

This paper investigates the existence of perfect codes in generalized Fibonacci cubes, extending previous non-existence results and identifying specific conditions under which perfect codes do exist.

Contribution

The authors prove the existence of perfect codes in generalized Fibonacci cubes for certain dimensions and parameters, advancing understanding of coding structures in these graphs.

Findings

Perfect codes exist in mma_n(1^s) for n=2^p-1 and s  3.2^{p-2}

Non-existence of perfect codes in standard Fibonacci cubes for n  4

Extension of perfect code existence to a broader class of Fibonacci-like graphs

Abstract

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 $p\geq 2$.