The Icosian Code and the E8 Lattice: A New 4 × 4 Space-Time Code with Non-vanishing Determinant

Abstract

This paper introduces a new full-rate, full-diversity space-time code for 4 transmit antennas. 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 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 similar 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