论文标题
在几乎完美的线性Lee包装半径2
On almost perfect linear Lee codes of packing radius 2
论文作者
论文摘要
50多年前，Golomb and Welch猜想没有完美的Lee代码$ C $包装RADIUS $ R $ in $ \ MATHBB {Z}^{n} $ for $ r \ geq2 $和$ n \ geq 3 $。最近，Leung和第二作者证明，如果$ C $是线性的，那么Golomb-Welch的猜想对$ r = 2 $和$ n \ geq 3 $有效。在本文中，我们考虑了第二好的可能性的线性Lee代码分类，即$ \ Mathbb {Z}^n $的晶格包装密度，由Lee Spheres $ s（n，r）$等于$ \ frac $ \ frac {| s（n，r）| s（n，r）| s（n，r）| s（n，r）{| s（n，r）| +1} $。我们表明，对于$ r = 2 $和$ n \ equiv 0,3,4 \ pmod {6} $，永远无法实现此包装密度。
More than 50 years ago, Golomb and Welch conjectured that there is no perfect Lee codes $C$ of packing radius $r$ in $\mathbb{Z}^{n}$ for $r\geq2$ and $n\geq 3$. Recently, Leung and the second author proved that if $C$ is linear, then the Golomb-Welch conjecture is valid for $r=2$ and $n\geq 3$. In this paper, we consider the classification of linear Lee codes with the second-best possibility, that is the density of the lattice packing of $\mathbb{Z}^n$ by Lee spheres $S(n,r)$ equals $\frac{|S(n,r)|}{|S(n,r)|+1}$. We show that, for $r=2$ and $n\equiv 0,3,4 \pmod{6}$, this packing density can never be achieved.
