By using totally isotropic subspaces in an orthogonal space Omega^{+}(2i,2),several infinite families of packings of 2^k-dimensional subspaces of real2^i-dimensional space are constructed, some of which are shown to be optimalpackings. A certain Clifford group underlies the construction and links thisproblem with Barnes-Wall lattices, Kerdock sets and quantum-error-correctingcodes.
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