Bases of minimal vectors in tame lattices

Damir, Mohamed Taoufiq; Mantilla-Soler, Guillermo

doi:10.48550/arxiv.2006.16794

Cited by 2 publications

(16 citation statements)

References 18 publications

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“…Motivated by these findings, tame lattices were introduced in [14], providing a source for constructing explicit well-rounded lattices. In the present paper, we take a step further and show that tame lattices also give rise to generic well-rounded lattices.…”

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confidence: 99%

“…• In Theorem 15, we state a stronger version of Theorem 4.9 in [14], namely that a well-rounded lattice of the form L (r,s) v1…”

mentioning

confidence: 99%

“…Since K is tame, d is also equal to the product of the primes that ramify in K. Based in this explicit description of the trace form over O K , in [15] the authors have constructed families of well-rounded lattices that are sublattices of O K . In [14], the notion of tame lattice is introduced by axiomatizing the key properties of the integral trace over degree p Galois extensions. For instance notice that that a and h satisfy a − h(p − 1) = 1 and a, h ≥ 0.…”

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confidence: 99%

“…For instance notice that that a and h satisfy a − h(p − 1) = 1 and a, h ≥ 0. In [14] these lattices are used to develop a procedure to construct well-rounded lattices, in fact strongly well-rounded, for which the previous results on Galois degree p number fields case is just a particular example. In the present paper we explore further the construction of tame lattices.…”

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confidence: 99%

“…In the present paper we explore further the construction of tame lattices. For instance, one of the key elements of the construction of strongly well-rounded lattices in [14] is the construction of a family of sublattices of the form L (r,s) Tv 1 ,v1 , see Definition 12 for details, for a given lattice L. Such construction is inspired by the study of the trace form over real number fields but its applications go beyond such fields. For instance we show that several lattices, including A n , D n , E 8 and their duals, all arise as special cases of this construction.…”

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Dense generic well-rounded lattices

Hollanti¹,

Mantilla-Soler²,

Miller³

2021

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It is well-known that the densest lattice sphere packings also typically have the largest kissing numbers. The sphere packing density maximization problem is known to have a solution among well-rounded lattices, of which the integer lattice Z n is the simplest example. The integer lattice is also an example of a generic well-rounded lattice, i.e., a well-rounded lattice with a minimal kissing number. However, the integer lattice has the worst density among well-rounded lattices. In this paper, the problem of constructing explicit generic well-rounded lattices with dense sphere packings is considered. To this end, so-called tame lattices recently introduced by Damir et al. are utilized. Tame lattices came to be as a generalization of the ring of integers of certain abelian number fields. The sublattices of tame lattices constructed in this paper are shown to always result in either a generic well-rounded lattice or the lattice An, with density ranging between that of Z n and An. Further, explicit deformations of some known densest lattice packings are constructed, yielding a family of generic well-rounded lattices with densities arbitrarily close to the optimum. In addition to being an interesting mathematical problem on its own, the constructions are also motivated from a more practical point of view. Namely, the constructed lattices provide good candidates for lattice codes used in secure wireless communications.

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confidence: 99%

“…• In Theorem 15, we state a stronger version of Theorem 4.9 in [14], namely that a well-rounded lattice of the form L (r,s) v1…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Dense generic well-rounded lattices

Hollanti¹,

Mantilla-Soler²,

Miller³

2021

Preprint

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Well-Rounded ideal lattices of cyclic cubic and quartic fields

Tran,

Le,

Tran

2023

Communications in Mathematics

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In this paper, we find criteria for when cyclic cubic and cyclic quartic fields have well-rounded ideal lattices. We show that every cyclic cubic field has at least one well-rounded ideal. We also prove that there exist families of cyclic quartic fields which have well-rounded ideals and explicitly construct their minimal bases. In addition, for a given prime number $p$, if a cyclic quartic field has a unique prime ideal above $p$, then we provide the necessary and sufficient conditions for that ideal to be well-rounded. Moreover, in cyclic quartic fields, we provide the prime decomposition of all odd prime numbers and construct an explicit integral basis for every prime ideal.

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Bases of minimal vectors in tame lattices

Cited by 2 publications

References 18 publications

Dense generic well-rounded lattices

Dense generic well-rounded lattices

Well-Rounded ideal lattices of cyclic cubic and quartic fields

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