论文标题
卢卡斯立方体中（非）存在完美代码
The (non-)existence of perfect codes in Lucas cubes
论文作者
论文摘要
尺寸n的斐波那契立方体（称为$γ$ n）是由顶点诱导的n-Cube 5 q n的子图，没有连续的1。 Ashrafi和他的合着者证明了N $ \ ge $ 4的$γ$ n中完美代码的不存在。作为一个空旷的问题，作者建议考虑在纤维纤维数据概括中存在完美的代码。最直接的概括是由没有1 s的字符串引起的$γ$ n（1 s）的子图，其中s $ \ ge $ 2是给定整数。在先例的工作10中，我们证明了$γ$ n（1 s）的完美代码，n = 2 p - 1和s $ \ ge $ 3.2 p $ \ ge $ 2 $ 2。在本说明中，我们证明了$λ$ n的完美代码不存在，n $ \ ge $ 4和15证明了某些广义的卢卡斯立方体$λ$ n（1 s）中存在完美代码。
The Fibonacci cube of dimension n, denoted as $Γ$ n , is the subgraph of the n-cube 5 Q n induced by vertices with no consecutive 1's. Ashrafi and his co-authors proved the non-existence of perfect codes in $Γ$ n for n $\ge$ 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 $Γ$ n (1 s) of subgraphs induced by strings without 1 s as a substring where s $\ge$ 2 is a given integer. In a precedent work 10 we proved the existence of a perfect code in $Γ$ n (1 s) for n = 2 p -- 1 and s $\ge$ 3.2 p--2 for any integer p $\ge$ 2. The Lucas cube $Λ$ n is obtained from $Γ$ n 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 $Λ$ n for n $\ge$ 4 and 15 prove the existence of perfect codes in some generalized Lucas cube $Λ$ n (1 s).
