A Construction of Rotated Lattices via Totally Real Subfields of the Cyclotomic Field $\mathbb{Q}({\zeta }_p)$

ABSTRACT

The theory of lattices have shown to be useful in information theory and rotated lattices with high modulation diversity have been extensively studied as an alternative approach for transmission over a Rayleigh-fading channel, where the performance of this modulation schemes essentially depends on the modulation diversity and on the minimum product distance to achieve substantial coding gains. The maximum diversity of a rotated lattice is guaranteed when we use totally real number fields and the minimum product distance is optimized by considering fields with minimum discriminant. In this paper, we present construction of a full diversity rotated lattice for the Rayleigh fading channel in Euclidean space with full diversity, where this construction is through a totally real subfield $𝕂$ of the cyclotomic field $$\mathbb{Q}(\zeta p)$$, where $p$ is an odd prime, obtained by endowing their ring of integers.

Key words
lattices; cyclotomic fields; algebraic number field; rotated lattice