The integral trace form of cyclic extensions of odd prime degree

Oliveira, Everton Luiz de; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega; Lopes, José Othon Dantas

doi:10.1216/rmj-2017-47-4-1075

Cited by 10 publications

(9 citation statements)

References 14 publications

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“…is a sub-lattice of O K with a minimal basis and minimum λ 1 = n(p−1)+m 2 p . When restricting this example to the case m ≡ 1 (mod p) the results [Theorem 4.1, [11]] and [Theorem 3.3, [12]] are recovered.…”

Section: Conditions On Minimal Basismentioning

confidence: 89%

“…Remark 1.1. Suppose that K is a Galois number field of prime degree p, unramified at p. If one uses the theorem above with r = 1, over the Lagrangian lattice O K , then the results of [11] and [12] are recovered.…”

Section: Suppose Thatmentioning

confidence: 99%

“…Suppose that p is an odd prime and that K is a degree p tame Galois number field. In [11] and [12] the authors give a set of conditions on positive integers m ≡ 1 (mod p) so that the sub-lattice of O K given by {x ∈ O K : Tr K/Q (x) ≡ 0 (mod m)} has a minimal basis. One of the key properties of O K behind such result is a Theorem of Conner and Perlis that shows that O K has a Lagrangian basis.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bases of minimal vectors in tame lattices

Damir¹,

Mantilla-Soler²

2020

Preprint

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Motivated by the ring of integers of cyclic number fields of prime degree, we introduce the notion of Lagrangian lattices. Furthermore, given an arbitrary non-trivial lattice L we construct a family of full-rank sub-lattices {Lα} of L such that whenever L is Lagrangian it can be easily checked whether or not Lα has a basis of minimal vectors. In this case, a basis of minimal vectors of Lα is given.

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Section: Conditions On Minimal Basismentioning

confidence: 89%

Section: Suppose Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bases of minimal vectors in tame lattices

Damir¹,

Mantilla-Soler²

2020

Preprint

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“…The next proposition, stated and proved in [19,Corollary 2.3], will be used in the next section. Before recalling it, we need a few assumptions.…”

Section: Trace Forms For Cyclotomic Fieldsmentioning

confidence: 99%

Trace forms of certain subfields of cyclotomic fields and applications

Ferrari

Andrade

Araujo

et al. 2020

Journal of Algebra Combinatorics Discrete Structures and Applications

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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 and Λ11 and they are images of 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 7 over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity.

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“…In [13], a lattice construction using cyclic number fields of degree p, where p is unramified, was presented. Lattices whose center densities are arbitrarily close to optimal values were obtained.…”

Section: Trace Form Over Cyclic Number Fields Of Degree Pmentioning

confidence: 99%

Constructions of Dense Lattices of Full Diversity

Andrade¹,

Ferrari²,

Interlando

et al. 2020

Tend. Mat. Apl. Comput.

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A lattice construction using Z-submodules of rings of integers of number ﬁelds is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number ﬁeld can be calculated by the trace form of the ﬁeld restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subﬁeld of a cyclotomic ﬁeld. Our focus is on totally real number ﬁelds since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

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The integral trace form of cyclic extensions of odd prime degree

Cited by 10 publications

References 14 publications

Bases of minimal vectors in tame lattices

Bases of minimal vectors in tame lattices

Trace forms of certain subfields of cyclotomic fields and applications

Constructions of Dense Lattices of Full Diversity

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