On the binary codes with parameters of doubly-shortened 1-perfect codes |
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Authors: | Denis S Krotov |
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Institution: | 1.Sobolev Institute of Mathematics,Novosibirsk,Russia;2.Mechanics and Mathematics Department,Novosibirsk State University,Novosibirsk,Russia |
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Abstract: | We show that any binary (n = 2
k
− 3, 2
n−k
, 3) code C
1 is a cell of an equitable partition (perfect coloring) (C
1, C
2, C
3, C
4) of the n-cube with the quotient matrix ((0, 1, n−1, 0)(1, 0, n−1, 0)(1, 1, n−4, 2)(0, 0, n−1, 1)). Now the possibility to lengthen the code C
1 to a 1-perfect code of length n + 2 is equivalent to the possibility to split the cell C
4 into two distance-3 codes or, equivalently, to the biparticity of the graph of distances 1 and 2 of C
4. In any case, C
1 is uniquely embedable in a twofold 1-perfect code of length n + 2 with some structural restrictions, where by a twofold 1-perfect code we mean that any vertex of the space is within radius
1 from exactly two codewords. By one example, we briefly discuss 2 − (n, 3, 2) multidesigns with similar restrictions. We also show a connection of the problem with the problem of completing latin
hypercuboids of order 4 to latin hypercubes. |
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Keywords: | |
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