论文标题
在延长的1完美bitrades上
On extended 1-perfect bitrades
论文作者
论文摘要
锤式方案中的$ 1 $完美代码$ h（n，q）$可以等效地定义为在任何坐标上穿刺后转向$ 1 $ - 完美的代码，因为与某些重量系数均匀地包装了某些重量的编码，因为与某些agipers相比，与某些重量的编码相比，与某些重量的编码相比，与某些$ 4 $ 4 $ 4相同。我们以五个不同的方式定义了$ h（n，q）$的扩展$ 1 $ - 完美的位拉德，对应于扩展$ 1 $ - 完美代码的不同定义，并证明了这些扩展$ 1 $ 1 $ - 完美比特的定义的等效性。对于$ q = 2^m $，我们证明当且仅当$ n = lq+2 $时，就存在这样的比特拉德。对于任何$ Q $，如果$ n $奇怪，我们证明不存在延长$ 1 $的位。 关键字：完美的代码，扩展完美的代码，Bitrade，完全常规的代码，均匀包装的代码。
Extended $1$-perfect codes in the Hamming scheme $H(n,q)$ can be equivalently defined as codes that turn to $1$-perfect codes after puncturing in any coordinate, as completely regular codes with certain intersection array, as uniformly packed codes with certain weight coefficients, as diameter perfect codes with respect to a certain anticode, as distance-$4$ codes with certain dual distances. We define extended $1$-perfect bitrades in $H(n,q)$ in five different manners, corresponding to the different definitions of extended $1$-perfect codes, and prove the equivalence of these definitions of extended $1$-perfect bitrades. For $q=2^m$, we prove that such bitrades exist if and only if $n=lq+2$. For any $q$, we prove the nonexistence of extended $1$-perfect bitrades if $n$ is odd. Keywords: Perfect code, Extended perfect code, Bitrade, Completely regular code, Uniformly packed code.
