Perfect and Related Codes

This dataset is devoted to 1-perfect codes. Currently, it is mainly focused on the concatenated ternary perfect codes, but there is also an additional content, see (3+) below, (1-2) This dataset contains all inequivalent concatenated ternary 1-perfect codes of length 13. Additionally, it contains some components necessary to obtain such concatenated codes, namely, collections of disjoint ternary distance-3 Reed-Muller-like codes of length 9, see p.(1) below. (3) Independently, check matrices of all (equivalent) binary Hamming codes of length 15 are collected in one file and all 232 inequivalent pairs of disjoint such codes are kept in another file. (4) The file 3ary-perfect-fullrank.zip contains examples of ternary 1-perfect codes of length 13, rank 13, and kernel dimension from 3 to 7.

In details, the dataset keeps:

(1) Inequivalent collections of (from 1 to 9) ternary $(9,3^6,3)_3$ codes that are subsets of the all-parity-check $(9,3^8,2)_3$ 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].

(2) All 93241327 inequivalent concatenated 1-perfect $(13,3^{10},3)_3$ codes (one code of rank 10, the Hamming code, 1164330 codes of rank 11, and 92076996 codes of rank 12). Each such code is the concatenation of one of the partitions of the all-parity-check $(9,3^8,2)_3$ code into $9$ codes, see p.(1), and one of the two inequivalent partitions of $H(4,3)$ into $1$-perfect codes (partition "A" consists of the translations of the same Hamming code, partition "B" is the other one). The $i$th code of a partition of length $9$ is concatenated with the $p(i)$th code of the partition "A" or "B" of length $4$, where $p$ is a permutation of $(0,1,2,3,4,5,6,7,8)$, and the resulting code is the union over all $i=0,1,2,3,4,5,6,7,8$.

(3) The file allHamming15.txt in the archive all_hamming15.zip contains the check matrices of all 64864800 binary Hamming codes of length 15 ( equivalently 64864800 projective geometries PG(3,GF(2)), or linear STS(15) ). The file Hamming_pairs1.txt in the same archive keeps all 232 nonequivalent ordered pairs of disjoint binary Hamming codes of length 15 (one code is linear, the other is a translation of a linear code). The file Hamming_pairs0.txt in the same archive keeps all 3374 ordered pairs of intersecting binary Hamming codes of length 15. Also, all 214970 nonequivalent ordered pairs of (15,256,5) Nordstrom-Robinson codes are collected in three files.

(4) The file 3ary-perfect-rank-kernel.zip contains examples of ternary 1-perfect codes of length 13 (i.e.,  (13,59049,3)_3 codes), rank 13 (i.e., full-rank), and kernel dimension from 3 to 7 and examples of concatenated (13,59049,3)_3 codes of different ranks and kernel dimensions. Each code is kept in a separate file where the codewords are listed in the ternary-vector form.

(5) The archiv equitable_partitions.zip contains the classification up to equivalence of some equitable partitions of hypercubes, including partitions of H(7,2) into multifold 1-perfect codes, orthogonal arrays OA(1536,13,2,7), OA(768,12,2,6), OA(1024,12,2,7), OA(512,11,2,6), simple orthogonal arrays OA(1536,12,2,7), OA(1792,12,2,7), balanced 5-correlation-immune 4-valued functions in 9 Boolean arguments.