An enumeration of 1-perfect ternary codes

10/12/2021

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by   Minjia Shi, et al.

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We study codes with parameters of the ternary Hamming (n,3^n-m,3) code, i.e., ternary 1-perfect codes. The rank of the code is defined as 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.