Eisenstein Integers Research Articles

Zero Divisor Graph for the Ring of Eisenstein Integers Modulo n

Let En be the ring of Eisenstein integers modulo n. In this paper we study the zero divisor graph Γ(En). We find the diameters and girths for such zero divisor graphs and characterize n for which the graph Γ(En) is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal.

Minimal covers of equivelar toroidal maps

Given any equivelar map on the torus, it is natural to consider its covering maps. The most basic of these coverings are finite toroidal maps or infinite tessellations of the Euclidean plane. In this paper, we prove that each equivelar map on the torus has a unique minimal toroidal rotary cover and also a unique minimal toroidal regular cover. That is to say, of all the toroidal rotary (or regular) maps covering a given map, there is a unique smallest. Furthermore, using the Gaussian and Eisenstein integers, we construct these covers explicitly.

Finding factors of factor rings over Eisenstein integers

Finding factors of factor rings over Eisenstein integers

Dimension of some fractal sets on hexagonal lattices

In this paper fractal dimension of fundamental domain boundary for all possible ternary quasicanonical numerical system in the ring of Eisenstein integers is calculated. Modified method that was used by W. Gilbert and J. Thuswaldner for computing of fundamental domain boundary fractal dimension for canonical numerical system is considered.

Generators for modules of vector-valued Picard modular forms

AbstractWe construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2, 1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.

Generators for modules of vector-valued Picard modular forms

AbstractWe construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2, 1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.

Frobenius circulant graphs of valency six, Eisenstein–Jacobi networks, and hexagonal meshes

A Frobenius group is a transitive but not regular permutation group such that only the identity element can fix two points. A finite Frobenius group can be expressed as G=K⋊H with K a nilpotent normal subgroup. A first-kind G-Frobenius graph is a Cayley graph on K with connection set S an H-orbit on K generating K, where H is of even order or S consists of involutions. We classify all 6-valent first-kind Frobenius circulant graphs such that the underlying kernel K is cyclic. We give optimal gossiping and routing algorithms for such a circulant and compute its forwarding indices, Wiener indices and minimum gossip time. We also prove that its broadcasting time is equal to its diameter plus two or three. We prove that all 6-valent first-kind Frobenius circulants with cyclic kernels are Eisenstein–Jacobi graphs, the latter being Cayley graphs on quotient rings of the ring of Eisenstein–Jacobi integers. We also prove that larger Eisenstein–Jacobi graphs can be constructed from smaller ones as topological covers, and a similar result holds for 6-valent first-kind Frobenius circulants. As a corollary any Eisenstein–Jacobi graph with order congruent to 1 modulo 6 and underlying Eisenstein–Jacobi integer not an associate of a real integer, is a cover of a 6-valent first-kind Frobenius circulant. A distributed real-time computing architecture known as HARTS or hexagonal mesh is a special 6-valent first-kind Frobenius circulant.

Quasi-Perfect Codes From Cayley Graphs Over Integer Rings

The problem of searching for perfect codes has attracted great attention since the paper by Golomb and Welch, in which the existence of these codes over Lee metric spaces was considered. Since perfect codes are not very common, the problem of searching for quasi-perfect codes is also of great interest. In this aspect, also quasi-perfect Lee codes have been considered for 2-D and 3-D Lee metric spaces. In this paper, constructive methods for obtaining quasi-perfect codes over metric spaces modeled by means of Gaussian and Eisenstein-Jacobi integers are given. The obtained codes form ideals of the integer ring thus preserving the property of being geometrically uniform codes. Moreover, they are able to correct more error patterns than the perfect codes which may properly be used in asymmetric channels. Therefore, the results in this paper complement the constructions of perfect codes previously done for the same integer rings. Finally, decoding algorithms for the quasi-perfect codes obtained in this paper are provided and the relationship of the codes and the Lee metric ones is investigated.

ETRU: NTRU over the Eisenstein integers

NTRU is a public-key cryptosystem based on polynomial rings over $$\mathbb Z .$$ Z . Replacing $$\mathbb Z $$ Z with the ring of Eisenstein integers yields ETRU. We prove through both theory and implementation that ETRU is faster and has smaller keys for the same or better level of security than does NTRU.

Lattice Network Codes Based on Eisenstein Integers

In this paper, we investigate lattice network codes (LNCs) constructed from Eisenstein integer based lattices. Quantization and encoding algorithms over Eisenstein integers are first introduced. Then, a union bound estimation (UBE) of the decoding error probability is derived when the shaping region of the LNC is a product of regular hexagons. Next, the Gaussian reduction algorithm is generalized to be applicable to complex lattices over Eisenstein integers such that an optimal coefficient vector can be found in the two-transmitter single-relay system. Based on the UBE, design criteria for optimal LNCs with minimum decoding error probability are formulated and applied to construct both Gaussian integer and Eisenstein integer based good LNCs from rate-1/2 feed-forward convolutional codes by Complex Construction A. The constructed codes provide up to 7.65 dB nominal coding gains over Rayleigh fading channels. Furthermore, we introduce the construction of LNCs from linear codes by Complex Construction B. The nominal coding gains and error performance of the LNCs thus constructed are explicitly analyzed. Examples show that the LNCs constructed by Complex Construction B provide a better tradeoff between code rate and nominal coding gain.

On the Classification of Lattices Over Which Are Even Unimodular -Lattices of Rank 32

We classify the lattices of rank 16 over the Eisenstein integers which are even unimodular -lattices (of dimension 32). There are exactly 80 unitary isometry classes.

ON THE FACTOR RINGS OF EISENSTEIN INTEGERS

In this study, we will answer the question “What can we say about the factor rings of Eisenstein integers which arise naturally when we consider the factor rings of the ring of integers which is the fundamental concept of algebra. In other words, we will characterize the structure of factor rings for the ring of Eisenstein integers.

WELL-ROUNDED SUBLATTICES

Given a lattice in the plane, we consider zeta-functions encoding the number of well-rounded sublattices of a given index. We are particularly interested in the abscissa of convergence of this function and show that the quality of convergence is related to arithmeticity questions concerning the ambient lattice. In particular, we discover that there are infinitely many similarity classes of well-rounded sublattices in a plane lattice if there is at least one. This generalizes results about the rings of Gaussian and of Eisenstein integers by Fukshansky and his coauthors.

Complex parametrization of triangulations on oriented maps

We consider here triangulations of oriented maps having a specified set S of vertices of degree different from 6 and some other vertices of degree 6. Such map can be described by specifying the relative positions between elements of S using Eisenstein integers. We first consider the case of 1 parameter, which corresponds to the Goldberg-Coxeter construction. Then we develop the general theory, the special case of positive curvature studied by Thurston and finally extend the theory to quadrangulations and some other cases. In the last section we expose application of parameterizations to the study of zigzags.

On the distribution of cubic exponential sums

Abstract. Using the theory of metaplectic forms, we study the asymptotic behavior of cubic exponential sums over the ring of Eisenstein integers. In the first part of the paper, some non-trivial estimates on average over arithmetic progressions are obtained. In the second part of the paper, we prove that the sign of cubic exponential sums changes infinitely often, as the modulus runs over almost prime integers.

The Topology of Gaussian and Eisenstein-Jacobi Interconnection Networks

Earlier authors have used quotient rings of Gaussian and Eisenstein-Jacobi integers to construct interconnection networks with good topological properties. In this paper, we present a unified study of these two types of networks. Our results include decomposing the edges into disjoint Hamiltonian cycles, a simplification of the calculation of the Eisenstein-Jacobi distance, a distribution of the distances between Eisenstein-Jacobi nodes, and shortest path routing algorithms. In particular, the known Gaussian routing algorithm is simplified.

NTRU over rings beyond $${\mathbb{Z}}$$

The NTRU cryptosystem is constructed on the base ring $${\mathbb{Z}}$$ . We give suitability conditions on rings to serve as alternate base rings. We present an example of an NTRU-like cryptosystem based on the Eisenstein integers $${\mathbb{Z}[\zeta_3]}$$ , which has a denser lattice structure than $${\mathbb{Z}}$$ for the same dimension, and which furthermore presents a more difficult lattice problem for lattice attacks, for the same level of decryption failure security.

On the [formula omitted] complex reflection group

We give a computer-free proof of a theorem of Basak, describing the group generated by 16 complex reflections of order 3, satisfying the braid and commutation relations of the Y 555 diagram. The group is the full isometry group of a certain lattice of signature ( 13 , 1 ) over the Eisenstein integers Z [ 1 3 ] . Along the way we enumerate the cusps of this lattice and classify the root and Niemeier lattices over Z [ 1 3 ] .

Self-similar hex-sums of squares

Self-Similar Hex-Sums of Squares R. BREU ~ eading the very stimulating article by A. van der Poorten, et al. in the Mathematical Intelligencer [1], I realized that self-similar sums of squares have ~. ~ , a nice analogue in what I call self-similar hex-sums of squares, just as decomposing primes of the form 4q + 1 into sums of squares a 2 + b 2 o r factoring 4q + 1 in the ring of Gaussian integers has an analogue in the ring of Eisenstein integers, that is, decomposing primes of the form 3q + 1 into "hex-sums" of squares a 2 + b 2 a b . ("Hex," as the fundamental domain in the Eisenstein lattice is a hexagon, versus a square in the Gaussian case.) Eisenstein integers have the form a -+ b~0, where ~o = ( 1 + X/-Z-3)/2. Their n o r m a 2 -4b 2 + ab corresponds to the n o r m a 2 + b 2 o f Gaussian integers. Both signs can be chosen, as the "upper sign" version is just the mirrored version of the "lower sign," but then have to be used consistently throughout, that is, always the lower or always the upper sign. A norm equation of Gaussian integers, say (a 2 + b 2) (c 2 + d 2) = (ac + bd) 2 + (ad b c ) 2, translates into a similar norm equation of Eisenstein integers

Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers

All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in $\mathbb{Q}(\sqrt{-3})$ are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3.