We study codes with parameters of the ternary Hamming $(n=(3^m-1)/2,3^{n-m},3)$ code, i.e., ternary $1$-perfect codes. The rank of the code is defined to be the dimension of its affine span. We characterize ternary $1$-perfect codes of rank $n-m+1$, count their number, and prove that all such codes can be obtained from each other by a sequence of two-coordinate switchings. We enumerate ternary $1$-perfect codes of length $13$ obtained by concatenation from codes of lengths $9$ and $4$; we find that there are $93241327$ equivalence classes of such codes. Keywords: perfect codes, ternary codes, concatenation, switching.

翻译：我们研究具有三进制Hamming码参数$(n=(3^m-1)/2,3^{n-m},3)$的码，即三进制$1$-完美码。将码的秩定义为其仿射张成的维数。我们刻画了秩为$n-m+1$的三进制$1$-完美码，统计了其数量，并证明所有此类码可通过一系列两坐标切换相互得到。我们枚举了通过将长度为$9$和$4$的码级联得到的长为$13$的三进制$1$-完美码，发现此类码共有$93241327$个等价类。关键词：完美码，三进制码，级联，切换。