On additive generalization of Voronoï’s theory to algebraic number fields

Leibak, Alar

doi:10.3176/phys.math.2005.4.01

Cited by 5 publications

(2 citation statements)

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“…Evaluating the O K coefficient vector a is crucial in understanding the performance limit of the computation rate. Our goal is to find one coefficient vector or multiple coefficient vectors minimizing the so-called additive Humbert form [39]…”

Section: Achievable Computation Ratementioning

confidence: 99%

Ring Compute-and-Forward Over Block-Fading Channels

Lyu

Campello

Ling

2019

IEEE Trans. Inform. Theory

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The Compute-and-Forward protocol in quasi-static channels normally employs lattice codes based on the rational integers Z, Gaussian integers Z [i] or Eisenstein integers Z [ω], while its extension to more general channels often assumes channel state information at transmitters (CSIT). In this paper, we propose a novel scheme for Compute-and-Forward in block-fading channels without CSIT, which is referred to as Ring Compute-and-Forward because the fading coefficients are quantized to the canonical embedding of a ring of algebraic integers. Thanks to the multiplicative closure of the algebraic lattices employed, a relay is able to decode an algebraic-integer linear combination of lattice codewords. We analyze its achievable computation rates and show it outperforms conventional Compute-and-Forward based on Zlattices. By investigating the effect of Diophantine approximation by algebraic conjugates, we prove that the degrees-of-freedom (DoF) of the optimized computation rate is n/L, where n is the number of blocks and L is the number of users.

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Section: Achievable Computation Ratementioning

confidence: 99%

Ring Compute-and-Forward Over Block-Fading Channels

Lyu

Campello

Ling

2019

IEEE Trans. Inform. Theory

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“…Proof. By theorem 6 in [5], and by definition of the algebraic Hermite's constant γ K , for any a ∈ K >>0 it holds that…”

Section: Introductionmentioning

confidence: 99%

An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally Real Number Fields

Porter¹,

Mendelsohn²

2022

Preprint

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Let K be a totally real number field of degree n over Q, with discriminant and regulator ∆ K , R K respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above byMoreover, if K is a unit reducible field, the number of classes of perfect unary forms is bound above by O(∆ K exp(2n log(n))).

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An Explicit Construction of Initial Perfect Quadratic Forms over Some Families of Totally Real Number Fields

Leibak

2008

Lecture Notes in Computer Science

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On additive generalization of Voronoï’s theory to algebraic number fields

Cited by 5 publications

References 0 publications

Ring Compute-and-Forward Over Block-Fading Channels

Ring Compute-and-Forward Over Block-Fading Channels

An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally Real Number Fields

An Explicit Construction of Initial Perfect Quadratic Forms over Some Families of Totally Real Number Fields

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