Gaussian Integers Research Articles

Counting Perron numbers by absolute value

AbstractWe count various classes of algebraic integers of fixed degree by their largest absolute value. The classes of integers considered include all algebraic integers, Perron numbers, totally real integers, and totally complex integers. We give qualitative and quantitative results concerning the distribution of Perron numbers, answering in part a question of Thurston [‘Entropy in dimension one’, to appear in the Proceedings of the Banff Conference on Frontiers in Complex Dynamics].

Generation of Long Perfect Gaussian Integer Sequences

Recently, the perfect Gaussian integer sequences have been widely used in modern wireless communication systems, such as code division multiple access and orthogonal frequency-division multiplexing systems. This letter presents two different methods to generate the long perfect Gaussian integer sequences with ideal periodic auto-correlation functions. The key idea of the proposed methods is to use a short perfect Gaussian integer sequence together with the polynomial or trace computation over an extension field to construct a family of the long perfect Gaussian integer sequences. The period of the resulting long sequences is not a multiple of that of the short sequence, which has not been investigated so far. Compared with the already existing methods, the proposed methods have three significant advantages that a single short perfect Gaussian integer sequence is employed, the long sequences consist of two distinct Gaussian integers, and their energy efficiency is monotone increasing.

A CDMA scheme based on perfect Gaussian integer sequences

Both m-sequences and Gold sequences have been applied as the pseudonoise (PN) code in the code division multiple access (CDMA) scheme to enable distinct users to simultaneously share the available capacity of a common channel, which is called direct sequence CDMA (DS-CDMA). However, this scheme does not provide the required ideal periodic cross-correlation function property among various members of sequences within the same set. This paper proposes a new perfect Gaussian integer sequence (PGIS)-based CDMA scheme called PGIS-CDMA; in this scheme, the PN code of DS-CDMA is replaced by a set of PGISs. The intrinsic orthogonal property of PGIS can be applied to a CDMA system to achieve perfect cochannel interference separability under the ideal synchronism; it can also be adapted at the receiver to obtain diversity gain and improve the carry-to-interference ratio. These two concepts are further developed to implement two PGIS-CDMA configurations, and they can outperform DS-CDMA on the basis of Gold sequences that possess a favorable cross-correlation property for the CDMA scheme.

Pythagorean Triples, Complex Numbers, and Perplex Numbers

SummaryAfter reviewing the use of Gaussian integers to generate Pythagorean triples, we consider the history of the perplex numbers and use them in an analogous way to generate Pythagorean triples. We also show that primitive Pythagorean triples form a group under the perplex number system, different from the group determined by the complex number representations.

Packings of Equal Disks in a Square Torus

Packings of equal disks in the plane are known to have density at most $$\pi /\sqrt{12}$$ , although this density is never achieved in the square torus, which is what we call the plane modulo the square lattice. We find packings of disks in a square torus that we conjecture to be the most dense for certain numbers of packing disks, using continued fractions to approximate $$1/\sqrt{3}$$ and $$2-\sqrt{3}$$ . We also define a constant to measure the efficiency of a packing motived by a related constant due to Markov for continued fractions. One idea is to use the unique factorization property of Gaussian integers to prove that there is an upper bound for the Markov constant for grid-like packings. By way of contrast, we show that an upper bound by Gruber [In many cases optimal configurations are almost regular hexagonal, vol. 65, pp. 121–145, 1999; Geom Dedicata 84(1–3):271–320, 2001] for the error for the limiting density of a packing of equal disks in a planar square, which is on the order of $$1/\sqrt{N}$$ , is the best possible, whereas for our examples for the square torus, the error for the limiting density is on the order of 1 / N, where N is the number of packing disks.

Constructions of Gaussian Integer Periodic Complementary Sequences with ZCZ

This letter presents two constructions of Gaussian integer Z-periodic complementary sequences (ZPCSs), which can be used in multi-carriers code division multiple access (MC-CDMA) systems to remove interference and increase transmission rate. Construction I employs periodic complementary sequences (PCSs) as the original sequences to construct ZPCSs, the parameters of which can achieve the theoretical bound if the original PCS set is optimal. Construction II proposes a construction for yielding Gaussian integer orthogonal matrices, then the methods of zero padding and modulation are implemented on the Gaussian integer orthogonal matrix. The result Gaussian integer ZPCS sets are optimal and with flexible choices of parameters.

New Perfect Gaussian Integer Sequences from Cyclic Difference Sets

New Perfect Gaussian Integer Sequences from Cyclic Difference Sets

Divisor problem in special sets of Gaussian integers

Let $A_{1}$ and $A_{2}$ be fixed sets of gaussian integers. We denote by $\tau_{A_{1}, A_{2}}(\omega)$ the number of representations of $\omega$ in form $\omega=\alpha\beta$, where $\alpha \in A_{1}, \beta \in A_{2}$. We construct the asymptotical formula for summatory function $\tau_{A_{1}, A_{2}}(\omega)$ in case, when $\omega$ lie in the arithmetic progression, $A_{1}$ is a fixed sector of complex plane, $A_{2}=\mathbb{Z}[i]$.

A New Construction of Zero Correlation Zone Gaussian Integer Sequence Sets

In this letter, a construction of zero correlation zone (ZCZ) sequence sets over Gaussian integers is presented. Based on the characteristic sets of binary pseudo-random sequences and shift sequences, ZCZ Gaussian integer sequence sets with optimal or almost optimal parameters are constructed. Moreover, the ZCZ lengths of resultant ZCZ Gaussian integer sequence sets are flexible.

On automatic subsets of the Gaussian integers

Suppose that a and b are multiplicatively independent Gaussian integers, that are both of modulus ≥5. We prove that there exists a X⊂Z[i] which is a-automatic but not b-automatic. This settles a problem of Allouche, Cateland, Gilbert, Peitgen, Shallit, and Skordev.

Efficient Integer Coefficient Search for Compute-and-Forward

Integer coefficient selection is an important decoding step in the implementation of compute-and-forward (C-F) relaying scheme. Choosing the optimal integer coefficients in C-F has been shown to be a shortest vector problem (SVP) which is known to be NP hard in its general form. Exhaustive search of the integer coefficients is only feasible in complexity for small number of users while approximation algorithms such as Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm only find a vector within an exponential factor of the shortest vector. An optimal deterministic algorithm was proposed for C-F by Sahraei and Gastpar specifically for the real valued channel case. In this paper, we adapt their idea to the complex valued channel and propose an efficient search algorithm to find the optimal integer coefficient vectors over the ring of Gaussian integers and the ring of Eisenstein integers. A second algorithm is then proposed that generalises our search algorithm to the Integer-Forcing MIMO C-F receiver. Performance and efficiency of the proposed algorithms are evaluated through simulations and theoretical analysis.

On a functional equation on the set of Gaussian integers

On a functional equation on the set of Gaussian integers

Simultaneous p-orderings and minimizing volumes in number fields

In [VP], V.V. Volkov and F.V. Petrov consider the problem of existence of the so-called n-universal sets (related to simultaneous p-orderings of Bhargava) in the ring of Gaussian integers. A related problem concerning Newton sequences was considered by D. Adam and P.-J. Cahen in [AC]. We extend their results to arbitrary imaginary quadratic number fields and prove an existence theorem that provides a strong counterexample to a conjecture of Volkov–Petrov on minimal cardinality of n-universal sets. Along the way, we discover a link with Euler–Kronecker constants and prove a lower bound on Euler–Kronecker constants which is of the same order of magnitude as the one obtained by Ihara.

The Cayley Graphs Associated With Some Quasi-Perfect Lee Codes Are Ramanujan Graphs

Let $\Z_n[i]$ be the ring of Gaussian integers modulo a positive integer $n$. Very recently, Camarero and Mart\'{i}nez [IEEE Trans. Inform. Theory, {\bf 62} (2016), 1183--1192], showed that for every prime number $p>5$ such that $p\equiv \pm 5 \pmod{12}$, the Cayley graph $\mathcal{G}_p=\textnormal{Cay}(\Z_p[i], S_2)$, where $S_2$ is the set of units of $\Z_p[i]$, induces a 2-quasi-perfect Lee code over $\Z_p^m$, where $m=2\lfloor \frac{p}{4}\rfloor$. They also conjectured that $\mathcal{G}_p$ is a Ramanujan graph for every prime $p$ such that $p\equiv 3 \pmod{4}$. In this paper, we solve this conjecture. Our main tools are Deligne's bound from 1977 for estimating a particular kind of trigonometric sum and a result of Lov\'{a}sz from 1975 (or of Babai from 1979) which gives the eigenvalues of Cayley graphs of finite Abelian groups. Our proof techniques may motivate more work in the interactions between spectral graph theory, character theory, and coding theory, and may provide new ideas towards the famous Golomb--Welch conjecture on the existence of perfect Lee codes.

Families of Gaussian integer sequences with high energy efficiency

This study extends the authors’ earlier work to show that the Gaussian integer sequences of period p m − 1 with p − 2 non-zero out-of-phase autocorrelation values can be constructed from the known families of two-tuple-balanced p-ary sequences over the finite field 𝔽 p m , where p is an odd prime and m ≥ 2. The proposed Gaussian integer sequences have high energy efficiency and are superior to the perfect Gaussian integer sequences (introduced by Hu et al. in 2012) for the peak-to-average power ratio reduction in orthogonal frequency-division multiplexing systems.

Further results on degree‐2 perfect Gaussian integer sequences

A complex number whose real and imaginary parts are both integers is called a Gaussian integer. A Gaussian integer sequence is said to be perfect if it has an ideal periodic autocorrelation function (PACF) where all out-of-phase values are zero. Further, the degree of a Gaussian integer sequence is defined as the number of distinct non-zero Gaussian integers within one period of the sequence. Recently, the perfect Gaussian integer sequences have been found important practical applications as signal processing tools for orthogonal frequency-division multiplexing systems. The present article generalises the authors’ earlier paper by Lee et al. (2015) related to the Gaussian integer sequences with ideal PACFs. By the applications of two-tuple-balanced binary sequences and cyclic difference sets, a number of new degree-2 perfect Gaussian integer sequences with different periods are obtained.

Improved lattice reduction‐aided linear precoding in MIMO systems with limited feedback

An approach of scaling and shifting constellation for MIMO lattice reduction(LR)-aided precoding is proposed. The proposed scheme is beneficial to improve the performance in terms of both BER and ergodic capacity. The feedback is a Gaussian integer matrix in which the entries are preserved in a region with respect to modulo operation. Then, the amount of feedback information is reduced significantly. Simulation results show that the LR-aided precoding scheme using the improved version of the constellation apparently improves system performance.

Prime labeling of families of trees with Gaussian integers

A graph on n vertices is said to admit a prime labeling if we can label its vertices with the first n natural numbers such that any two adjacent vertices have relatively prime labels. Here we extend the idea of prime labeling to the Gaussian integers, which are the complex numbers whose real and imaginary parts are both integers. We begin by defining an order on the Gaussian integers that lie in the first quadrant. Using this ordering, we show that several families of trees admit a prime labeling with the Gaussian integers.

Representation of a class of multiple fractional part integrals and their closed form

ABSTRACTIn this paper, the following generalized multiple fractional part integrals: and are considered for complex numbers , positive integers and positive number a, where denotes the fractional part of u. Furthermore, and can be transformed into the calculation of definite fractional part integral and , which overcomes the shortcomings of low speed and poor accuracy of the multiple case. Moreover, some identities and recursive formulas of the above integrals are obtained.

Discrete Space-time and Lorentz Transformations

Abstract Alfred Schild established conditions where Lorentz transformations map world-vectors (ct, x, y, z) with integer coordinates onto vectors of the same kind. The problem was dealt with in the context of tensor and spinor calculus. Due to Schild’s number-theoretic arguments, the subject is also interesting when isolated from its physical background. Schild’s paper is not easy to understand. Therefore, we first present a streamlined version of his proof which is based on the use of null vectors. Then we present a purely algebraic proof that is somewhat shorter. Both proofs rely on the properties of Gaussian integers