1-Perfect Uniform and Distance Invariant Partitions |
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Authors: | J Rifà J Pujol J Borges |
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Affiliation: | (1) Dept. d'Informàtica, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain (e-mail: {jborges, jrifa, jpujol}@ccd.uab.es), ES |
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Abstract: | Let F
n
be the n-dimensional vector space over ℤ2. A (binary) 1-perfect partition of F
n
is a partition of F
n
into (binary) perfect single error-correcting codes or 1-perfect codes. We define two metric properties for 1-perfect partitions:
uniformity and distance invariance. Then we prove the equivalence between these properties and algebraic properties of the
code (the class containing the zero vector). In this way, we characterize 1-perfect partitions obtained using 1-perfect translation
invariant and not translation invariant propelinear codes. The search for examples of 1-perfect uniform but not distance invariant
partitions enabled us to deduce a non-Abelian propelinear group structure for any Hamming code of length greater than 7.
Received: March 6, 2000; revised version: November 30, 2000 |
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Keywords: | : Perfect propelinear codes Perfect uniform partitions Perfect distance invariant partitions |
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