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The classification of some perfect codes. (English) Zbl 1048.94018
Summary: Perfect 1-error correcting codes $$C$$ in $$\mathbb{Z}_2^n$$, where $$n=2^m-1$$, are considered. Let $$\langle C\rangle$$ denote the linear span of the words of $$C$$, and let the rank of $$C$$ be the dimension of the vector space $$\langle C\rangle$$. It is shown that if the rank of $$C$$ is $$n-m+2$$ then $$C$$ is equivalent to a code given by a construction of K. Phelps [SIAM J. Algebraic Discrete Methods 5, 224–228 (1984; Zbl 0546.94015)]. These codes are, in case of rank $$n-m+2$$, described by a Hamming code $$H$$ and a set of MDS-codes $$D_h$$, $$h \in H$$, over an alphabet with four symbols. The case of rank $$n-m+1$$ is much simpler: Any such code is a Vasil’ev code.

##### MSC:
 94B25 Combinatorial codes
perfect codes
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