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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
Keywords:
perfect codes
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