The Icosian Code and the $E_8$ Lattice: A New $4\,\times\,4$ Space–Time Code With Nonvanishing Determinant

This paper introduces a new rate-2, full-diversity space-time code for four transmit antennas and one receive antenna. The 4times4 codeword matrix consists of four 2times2 Alamouti blocks with entries from
Q
(
i
,radic5) , and these blocks can be viewed as quaternions which in turn represent rotations in
R

³
. The Alamouti blocks that appear in a codeword are drawn from the icosian ring consisting of all linear combinations of 120 basic rotations corresponding to symmetries of the icosahedron. This algebraic structure is different from the Golden code, but the complex entries are taken from a common underlying field. The minimum determinant is bounded below by a constant that is independent of the signal constellation, and the new code admits a simple decoding scheme that makes use of a geometric correspondence between the icosian ring and the
E

₈
lattice.