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The codes and the lattices of Hadamard matrices
It has been observed by Assmus and Key as a result of the complete
classification of Hadamard matrices of order 24, that the extremality of the
binary code of a Hadamard matrix H of order 24 is equivalent to the extremality
of the ternary code of H^T. In this note, we present two proofs of this fact,
neither of which depends on the classification. One is a consequence of a more
general result on the minimum weight of the dual of the code of a Hadamard
matrix. The other relates the lattices obtained from the binary code and from
the ternary code. Both proofs are presented in greater generality to include
higher orders. In particular, the latter method is also used to show the
equivalence of (i) the extremality of the ternary code, (ii) the extremality of
the Z_4-code, and (iii) the extremality of a lattice obtained from a Hadamard
matrix of order 48.Comment: 16 pages. minor revisio
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