An enumeration of 1-perfect ternary codes

TL;DR

This paper characterizes, counts, and classifies ternary 1-perfect codes, especially those of a specific rank, and enumerates codes of length 13 constructed via concatenation, revealing over 93 million equivalence classes.

Contribution

It provides a complete characterization of certain ternary 1-perfect codes, counts their number, and demonstrates their construction through concatenation and switching techniques.

Findings

Characterized ternary 1-perfect codes of rank n-m+1.

Counted the number of such codes and proved their interconnectivity via switchings.

Enumerated over 93 million equivalence classes of length 13 codes from concatenation.

Abstract

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.