Trace Forms of Certain Subfields of Cyclotomic Fields and Applications

In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We also obtain a closed formula for the Gram matrix of algebraic lattices obtained from these subfields. The obtained lattices are rotated versions of the lattices Λ9,Λ10\Lambda_9, \Lambda_{10}Λ​9​​,Λ​10​​ and Λ11\Lambda_{11}Λ​11​​ and they are images of Z\mathbb{Z}Z-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. We also obtain algebraic lattices in odd dimensions up to 777 over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity

Characterization of Totally Real Subfields of 2-Power Cyclotomic Fields and Applications to Signal Set Design

A classification of all totally real subfields K of cyclotomic field Q(zeta_{2^r}), for any r ≥ 4, and the fully-diverse related versions of the Z^n-lattice are presented along with closed-form expressions for their minimum product distance. Any totally real subfield K of Q(zeta_{2^r}) must be of the form K=Q(zeta_{2^2} + zeta_{2^2}^{-1}), where s = r − j for some 0 ≤ j ≤ r − 3. Signal constellations for transmitting information over both Gaussian and Rayleigh fading channels (which can be useful for mobile communications) can be carved out of those lattices

Trace forms of certain subfields of cyclotomic fields and applications

In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We also obtain a closed formula for the Gram matrix of algebraic lattices obtained from these subfields. The obtained lattices are rotated versions of the lattices Λ9,Λ10\Lambda_9, \Lambda_{10}Λ​9​​,Λ​10​​ and Λ11\Lambda_{11}Λ​11​​ and they are images of Z\mathbb{Z}Z-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. We also obtain algebraic lattices in odd dimensions up to 777 over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity

Two matrix-based lattice construction techniques

Let m and n be integers greater than 1. Given lattices A and B of dimensions m and n, respectively, a technique for constructing a lattice from them of dimension m+n-1 is introduced. Furthermore, if A and B possess bases satisfying certain conditions, then a second technique yields a lattice of dimension m+n-2. The relevant parameters of the new lattices are given in terms of the respective parameters of A,B, and a lattice C isometric to a sublattice of A and B. Denser sphere packings than previously known ones in dimensions 52, 68, 84, 248, 520, and 4098 are obtained. © 2012 Elsevier Inc. All rights reserved