Liu, J; Calderbank, AR, *The Icosian Code and the $E_8$ Lattice: A New $4\,\times\,4$ Space–Time Code With Nonvanishing Determinant*,
Ieee Transactions on Information Theory, vol. 54 no. 8
(August, 2008),
pp. 3782-3789 [doi] .
**Abstract:**

*This paper introduces a new rate-2, full-diversity space-time code for four transmit antennas and one receive antenna. The 4 × 4 codeword matrix consists of four 2 × 2 Alamouti blocks with entries from Q(i,√5), and these blocks can be viewed as quaternions which in turn represent rotations in R3. 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 E8 lattice. © 2008 IEEE.*