Ternary Perfect and Related Codes

This dataset keeps inequivalent collections of (from 1 to 9) ternary $(9,3^6,3)_3$ codes that are subsets o the all-parity-check $(9,3^6,3)_3$ code. The equivalence is understood in the sense of the automorphisms of the Hamming graph $H(9,3)$. There are 4 equivalence classes of such codes; 141 equivalence classes of pairs of disjoint codes; 10956 equivalence classes of triples; 118388 classes for 4 disjoint codes; 501915 for 5; 945965 for 6; 755066 for 7; 314833 for 8; 65436 equivalence classes of partitions of the all-parity-check $(9,3^6,3)_3$ code into 9 distance-3 codes. Such partitions, in combination with partitions of the Hamming space $H(4,3)$ into 9 1-perfect codes (the two inequivalent partitions of $H(4,3)$ can also be found in the file H43.py in this dataset), can be used to construct 1-perfect ternary codes of length 13 by concatenation, see [Romanov, A. M. On Non-Full-Rank Perfect Codes Over Finite Fields. Des. Codes Cryptography, 2019, 87, 995-1003]. The next goal of this project is to classify all concatenated 1-perfect $(13,3^{10},3)_3$ codes up to equivalence. The dataset will be updated.