1-Perfect Uniform and Distance Invariant Partitions

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.