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Abstract
In this paper, we propose a construction of lattices in dimension 8n via octonion orders over a totally real number field. To show the potential of this construction, we present rotated versions of $E_8$, $\Lambda_{16}$, the densest known lattices in these dimensions and a lattice with the same density of the Barnes-Wall lattice in dimension 32. Perspectives applications of these lattices based on octonion algebras are in lattice-based cryptography and lattice coding for MIMO and MISO channels.
Highlights
Signal constellations having a lattice structure have been studied as meaningful tools for transmitting data over both Gaussian and single-antenna Rayleigh fading channels [1]
Algebraic number theory has been used as mathematical tool that enables the design of good coding schemes due to coding/decoding properties
It has been shown that algebraic lattices, i.e., lattices constructed via the canonical embedding of an algebraic number field, provide an efficient tool for designing lattice codes for transmission over the single-antenna Rayleigh fading channel [3]
Summary
Signal constellations having a lattice structure have been studied as meaningful tools for transmitting data over both Gaussian and single-antenna Rayleigh fading channels [1]. It has been shown that algebraic lattices, i.e., lattices constructed via the canonical embedding of an algebraic number field, provide an efficient tool for designing lattice codes for transmission over the single-antenna Rayleigh fading channel [3]. Having the construction of rotated lattices as our goal, we are interested in constructing dense lattices from octonion orders over a totally real number field. In this construction we use algebraic lattice theory.
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