Orthogonal designs, self-dual codes, and the leech lattice

Symmetric designs and Hadamard matrices are used to construct binary and ternary self-dual codes. Orthogonal designs are shown to be useful in construction of self-dual codes over large fields. In this paper, we first introduce a new array of order 12, which is suitable for any set of four amicable circulant matrices. We apply some orthogonal designs of order 12 to construct new self-dual codes over large finite fields, which lead us to the odd Leech lattice by Construction A. (c) 2005 Wiley Periodicals, Inc.