The Classification of Some Perfect Codes

AbstractPerfect 1-error correcting codes C in Z2n, where n=2m−1, are considered. Let $$ \left\langle C \right\rangle $$ ; denote the linear span of the words of C and let the rank of C be the dimension of the vector space $$ \left\langle C \right\rangle $$ . 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 Phelps. These codes are, in case of rank n−m+2, described by a Hamming code H and a set of MDS-codes Dh, 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.