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Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.


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Journal ArticleDOI
18 Jun 2018
TL;DR: Construction A of lattices is extended to the ring of algebraic integers of a general imaginary quadratic field that may not form a principal ideal domain (PID) and it is shown that such a construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization.
Abstract: In this paper, we extend Construction A of lattices to the ring of algebraic integers of a general imaginary quadratic field that may not form a principal ideal domain (PID). We show that such a construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. As an application, we then apply the proposed lattices to the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all PIDs. A novel scheme called adaptive compute-and-forward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive compute-and-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely.

16 citations

Journal ArticleDOI
TL;DR: In this article, necessary and sufficient conditions for representing certain classes of primes by given quadratic forms are found by generalizing techniques of rational number theory, and the main result is that if m = 5 or 13, and if p is a rational prime such that ( − 1 p ) = 1 = ( m p ), then a necessary and necessary condition that x 2 + 4 my 2 = p for some rational integers x and y is that [ ϵ m p ] = 1, where ϵm denotes the fundamental unit of the field Q(m 1

16 citations

Book ChapterDOI
28 Aug 1989
TL;DR: An algorithm is presented that when given algebraic numbers α1, α n and a parameter ɛ either finds an integer relation for α1,...,α n or proves that no relation of euclidean length shorter than 1/ɛ exists.
Abstract: A vector m=(m1,...,m n ) ∈ Zn \ {0} is called an integer relation for the real numbers α1,...,α n , if Σα i m i =0 holds. We present an algorithm that when given algebraic numbers α1,...,α n and a parameter ɛ either finds an integer relation for α1,...,α n or proves that no relation of euclidean length shorter than 1/ɛ exists. Each algebraic number is assumed to be given by its minimal polynomial and by a rational approximation precise enough to separate it from its conjugates.

16 citations

Journal ArticleDOI
TL;DR: An ''orbital Tutte polynomial'' associated with a dual pair M and M^* of matrices over a principal ideal domain R and a group G of automorphisms of the row spaces of the matrices is constructed.

16 citations

Journal ArticleDOI
TL;DR: In this article, the normal zeta functions of the Heisenberg groups H(R), where R is a compact discrete valuation ring of characteristic zero, were shown to satisfy functional equations upon inversion of the prime.
Abstract: We compute explicitly the normal zeta functions of the Heisenberg groups H(R), where R is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form H(O K ), where O K is the ring of integers of an arbitrary number field K, at the rational primes which are non-split in K. We show that these local zeta functions satisfy functional equations upon inversion of the prime.

16 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202250
2021117
2020121
2019111
201896