Some constructions of unimodular lattices via totally real subfields of the p-th cyclotomic field

Antonio Aparecido de Andrade; Grasiele C.  Jorge

doi:10.13069/jacodesmath.v13i2.349

Antonio Aparecido de Andrade São Paulo State University (UNESP), Institute of Biosciences, Humanities and Exact Sciences, São José do Rio Preto, Brazil
Grasiele C. Jorge Federal University of São Paulo (UNIFESP), Institute of Science and Technology, São José dos Campos, Brazil

DOI: https://doi.org/10.13069/jacodesmath.v13i2.349

Keywords: Rotated lattices, Ideal lattices, Full diversity lattices

Abstract

Algebraic number theory has recently attracted significant interest due to its role in algebraic lattice theory and in the design of codes for applications in coding theory. Algebraic lattices have been useful in information theory, where the problem of constructing lattices over number fields with full diversity and maximal minimum product distance has been investigated, since these parameters are directly related to error probabilities over Rayleigh fading channels. In this paper, we present a family of full diversity rotated unimodular lattices constructed via totally real subfields of the cyclotomic fields ℚ(ζ_p), with p an odd prime. A closed-form expression for the minimum product distance is derived.