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：我们研究了具有三进制汉明码参数（$n=(3^m-1)/2,3^{n-m},3$）的码，即三进制$1$-完备码。代码的秩被定义为其仿射空间的维数。我们表征了秩为$n-m+1$的三进制$1$-完备码，计算它们的数量，并证明所有这样的码可以通过一系列二维坐标切换从彼此获得。我们枚举了长度为$13$的三进制$1$-完备码，这些码是通过长度分别为$9$和$4$的码级联而获得的。我们发现这样的码有$93241327$个等价类。关键词：完美码，三进制码，级联，切换。
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