A lattice in $\mathbb R^n$ is a discrete subgroup of $\mathbb R^n$ or, equivalently, it is a subgroup of $\mathbb R^n$ generated by linearly independent vectors. Lattices have applications in geometric number theory, e.g. via Minkowski's theorem.

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Is this the subgroup lattice for $\Bbb{Z}_4 \times \Bbb{Z}_8$?

I have been attempting to create the subgroup lattice for $\Bbb{Z}_4 \times \Bbb{Z}_8$. I have, so far, this: http://www.scribd.com/doc/223680804/Subgroup-Lattice-of-Z-4-x-Z-8 While I have calculated ...
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203 views

How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
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Adelic lattices

Let $\mathbb{A} = \widehat{\mathbb{Z}} \otimes \mathbb{Q} \times \mathbb{R}$ be the adeles over $\mathbb{Q}$. In Deligne's article "Formes modulaires et representations de GL(2)" he states without ...
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The lattice points of a f.g. rational cone form a f.g. monoid.

In their book "Polytopes, Rings, and K-Theory" Bruns and Gubeladze sketch an alternative approach to Gordan's Lemma, which is stated in the headline (f.g. = finitely generated). I don't understand the ...
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Number of points in the diagonal of an $X \times Y$ square lattice box.

So assuming we have a $X \times Y$ lattice, Say for example a $3 \times 5$ like so * * * * * * * * * * * * * * * I need to find the number of points that each ...
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Nonplanar equilateral lattice “pentagons”

It is well-known that no two-dimensional point lattice contains a regular pentagon. (See for example http://mathworld.wolfram.com/LatticePolygon.html) The same is true for lattices in $\mathbb{R}^n$, ...
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continued fractions and convex hulls

If I remember correctly, there is a nice correspondence between continued fractions and convex hulls of lattice points in the plane. If $\theta>0$ is the slope of a line in $\mathbb{R}^2$ passing ...
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When does a simplex have an interior lattice point?

Given $r$ vectors $v_1, \dots, v_r$ in $\mathbb{Z}^n$, is there an easy way (in terms of the entries of the $v_i$) to determine if there is a point of $\mathbb{Z}^n$ in the interior of the simplex ...
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Lattice Reduction Problem: Minimizing the “Longest” Basis Vector

Suppose we have a basis for an integer lattice formed by the vectors $\vec v_1, \vec v_2, \ldots,\vec v_n$. Then let $A$ be the augmented matrix $( \vec v_1| \space \vec v_2| \cdots |\space \vec ...
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Term for intersection of lattice and convex region?

Is there a special term or convenient phrase for the restriction of a convex region to points of a lattice? This is motivated by wanting to talk about the feasible points of a discrete problem. I'd ...
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203 views

Probability distribution of a self avoiding walk

Preliminary: Consider a walk on the lattice $\mathbb{Z}_d$ lattice of length $N$. In a normal random walk, if we let $N$ get large the end position has a probability distribution (PDF) that looks ...
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Bijection between $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$ and lattices in $F^n$

I've come across mention of a bijection between lattices in $F^n$ ($F$ a field, in my case $\mathbb{C}(\!(t)\!)$) and elements of $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$, where $O$ is the ring ...
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Construction of a symplectic basis for a lattice

Let $(T,E)$ be a polarized abelian variety ($T=V/L$, $\dim_\mathbb{C} V=g$, $E:V\times V\to\mathbb{R}$ a nondegenerate real alternating bilinear form, with $E(L\times L)\subseteq\mathbb{Z}$ and ...
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The $n$-shortest lattice vectors problem in $\mathcal{R}^2$

I am looking for an algorithm to compute the $n$ shortest lattice vectors in $\mathcal{R}^2$. The problem statement is as follows: Given a lattice $L: \{ m \vec{u}+n\vec{v} \} \in \mathcal{R}^2$, a ...
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moduli of lattices

Consider the set $M$ of all (rank $g$) lattices in $g$-dimensional complex affine space $C^g$. Does M identify in some way with Siegel upper half space $H_g$? Let's say a lattice has CM if it has ...
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22 views

Which lattices are ideals of a number field?

Let $K$ be a number field, then its ring of integers $\mathcal{O}_K$ in the Minkowski space of $K$ is a lattice $\Lambda$. Is there some geometric descrpition/intuition that describes sublattices of ...
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Non zero Unimodular matrices

I have a question about the lattice reduction algorithms like LLL algorithm. Lattice reduction algorithms like LLL generate a unimodular matrix which makes more orthogonal basises for a given matrix. ...
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30 views

Is the Voronoi region of a lattice symmetric around origion?

Assume an n-dimensional lattice. Is the Voronoi region of the lattice symmetric around origion? In other word, is the following statement true? "if $x\in \mathcal{V}$ then $-x\in \mathcal{V}$" where ...
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42 views

Closest vector problem

Given is a vector $v=\begin{pmatrix}2,&-1,&0,&1\end{pmatrix}$ as the shortest vector of the lattice $\Lambda (B)$, where $B$ is determined as $B=\begin{pmatrix}4 &-3 & 2 & 0\\ ...
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55 views

Which matrices generate the same lattices?

Given are the following matrices: $A_{1}=\begin{bmatrix} 4 & 7 &9 \\ 3 & 5 &3 \\ 1 &2 &0 \end{bmatrix}$ , $A_{2}=\begin{bmatrix} 10 & 9 &6 \\ 11 & 7 &2 \\ ...
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integer programming with bounded dimension

We know that integer programming with bounded dimension or fixed number of variables can be solved in polynomial time by Lenstra's result(from results of the LLL algorithm). After heavy foraging i ...
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99 views

Lattice theory in mathematics and physics

I have undertaken a project examining lattice model and trying to construct algorithm that could work on all lattice (in physical sense, or crystal structure). I notice there is a branch in ...
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30 views

Inclusion-minimality of a lattice basis

An integer lattice is a subgroup of $\mathbb{Z}^n$. Since $\mathbb{Z}$ is PID, each lattice has a well-defined rank and a generating set of rank many elements is a basis. I wonder if there is a way ...
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A lattice-theoretic question related to noncommutative tori

[NCG] So I'm trying to pin down a fairly well-known bit of noncommutative-geometric folklore that says that for $\Theta \in M_N(\mathbb{Q})$ skew-symmetric, the corresponding noncommutative $N$-torus ...
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Why the following observations regarding lattices hold?

The following is an excerpt of a recent paper on lattice cryptography: Let $n$ and $q$ be integers [...], and let $\beta > 0$ . Given a uniformly random matrix $A \in \mathbb Z^{n \times m}_q$ ...
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Automorphisms of a lattice and changing to a nicer $\mathbb{Z}$-base

Suppose I have an integral lattice $L$ with an arbitrary $\mathbb{Z}$-base, equipped with a positive-definite nondegenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$, and an isometry $\nu$ ...
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128 views

solution count of quadratic form congruences

Let $(L,q)$ be an even Lorentzian lattice of signature $(1,l-1)$, i.e., $q(\lambda) \in \Bbb Z$ for all $\lambda \in L$ and the (non-degenerate) quadratic form $q$ is of type $(1,l-1)$ (the ...
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108 views

Norm of the generators of a fractional ideal.

Let $\mathcal{O}_l=\mathbb{Z}[\frac{1+\sqrt{-l}}{2}]$ with $l$ a prime number congruent to 3 mod 4. Let $\mathfrak{a}$ be a non-principal fractional ideal of $\mathcal{O}_l$. My questions are: Why ...
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301 views

How many amplitude levels does a squared QAM constellation have?

In the above question, I choose terminology from communication engineering, but I will precisely define all the involved quantities below. It is a geometric question and I am not sure if it is ...
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A variant of the “closest vector problem” (CVP) in lattice-based cryptography

Consider a public-key scheme on lattices, such as GGH. The private key is a basis $\mathbf{B} \in \mathbb{Z}^{m \times n}$ of a lattice $\mathcal{L}$ with good properties (such as short nearly ...
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Finding integer vectors in the column space of a matrix

Consider a given set $S \subset Z$. $S$ is a finite set. Matrix $A \in S^{N \times M}$ is also given. Does there exist an algorithm to find all the vectors belonging to the space Col$(A)\cap S^N$ ...
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smallest integer contained in sublattice $\Rightarrow$ $L'=[q,r\tau+s]$

Let $L'$ be a sublattice of the lattice $[1,\tau]$ in a imaginary quadratic field. Reminder: a lattice $L$ consists of the $\mathbb{Z}$-linear combinations of $1$ and $\tau$, with $\{1,\tau \}$ linear ...
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Automorphism group of torus fixing origin

I've got a short question: Suppose that you have some lattice $\Lambda$, say $\Lambda=\mathbb{Z}+\mathbb{Z}i$, and let $T$ be the torus $\mathbb{C}/\Lambda$, coming with the quotient map ...
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Upper bound on the product of norms of vectors in a lattice basis

The orthogonality defect of a lattice basis is the quotient of the product of the norms of the vectors in the basis and the determinant of the lattice. It is at least 1 by Hadamard's inequality. ...
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55 views

Integer Points on Circles

Let $r(n)$ denote the number of integral solutions to $a^2+b^2 = n$ where $a,b,n$ are integers. Furthermore, we count the pairs with regard to order and signs. (So if $(a,b)$ is a solution, so are ...
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Which of the following sets are sub lattices of $\mathbb{Z}^2$?

Here are the first three sets: $\{(x, y) \in \mathbb Z^2 : x + y = 1\}$. $\{(x, y) \in Z^2 : x + y = 0\} = S^2$. $\{(x,y) \in Z^2 :2\mid x\} = S^3$ I found that the first one is not a subgroup ...
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28 views

Covering a Disk in a Square Lattice: How Many Squares Are Needed?

I have a square lattice (the square side is $a$, and a disk centered at $(0,0)$ with radius 1. I want to know how many lattice squares are located completely inside the disk, and how many lattice ...
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99 views

Lattice orthogonal polyhedra face-area sequences: Golyhedra?

Let $P$ be a polyhedron, all of whose vertices are at points of $\mathbb{Z}^3$, all of whose edges are parallel to an axis, with every face simply connected, and the surface topologically a sphere. ...
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Lattice and their duals generate a full rank subgroup

Is there a lattice $L$ in $\mathbb{R}^2$ such that $L$ and $L^*$ together generate a rank 4 subgroup of $\mathbb{R}^2?$ If so, is it possible to do this if all the generators of $L$ and $L^*$ are ...
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15 views

Show transformation of lattice paths is a bijection

How do I show that the transformation (switching head and tail of a path) is indeed a bijection from lattice paths with excess k to lattice paths with excess k-1. I need to describe the inverse ...
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How is affine space analogue for lattices called?

Lattices are so like vector spaces that it seems natural to have an affine space construction for them. Unfortunately I could not find how such a construction is called. Could you please help me? ...
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Transforming FCC, BCC and HCP lattice types to cubes.

I was wondering if it is possible to transform the FCC, BCC and HCP into SC, or simple cubic lattices while preserving the lengths between the nodes? I would like to transform each into this: ...
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Ideals in cubic fields

I've been studying number fields, and the ideals of their integer rings, and I have a question. First, I know the following in the quadratic case. If a $\mathbb{Z}$-basis for the integer ring is ...
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Obstruction to construct the set of vectors in lattice

Lets consider a lattice $\mathbb{Z}^n$ with some unimodular scalar product $\mathbb{Z}^n \times \mathbb{Z}^n \mapsto \mathbb{Z}$ and the set of vectors $e_0,\ldots,e_k$ with conditions: $$ ...
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Its just one point… How do I find it?

Okay so here is the deal... I have a CLOSED convex polyhedron $Ax \le b$ (where $x$ is in $R^n$) and it has i vertices denoted $V_i$ such that $V_i = (x_{i1}, x_{i2}, \ldots, x_{iN})$ where $0 \le ...
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56 views

Determinant of a Lattice Code

For a binary $[n,k,d]$ code $C$ with lattice $\Lambda (C)$ I want to show that $det(\Lambda (C)) = 2^{n-2k}$. I'm having a little trouble with demonstrating this and it would help me very much ...
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Determinant of a superlattice

I'm beginning to study lattices and cannot solve the following exercise from the book: Let $\Lambda=\left\langle\mathbb{Z}^n,\left(\frac{a_1}q,\ldots,\frac{a_n}q\right)\right\rangle$, where ...
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Counting lattice points in a square region

We have a lattice which is spanned by the vectors $(a,b)$ and $(-b,a)$, where $a,b \in \mathbb{N}$. Now we have a square region centered at $(0,0)$ having sides of length $2r+1$ for $r\in \mathbb{N}$. ...
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60 views

Orthogonality in a lattice

Let $\Lambda$ be a lattice with a quadratic form $q$ of signature (3,19). Let $\Lambda_{\mathbb{R}}:=\Lambda\otimes \mathbb{R}$ and $W\subset \Lambda_{\mathbb{R}}$ a positive subspace of dimention 3. ...
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isomorphic lattices

I want to show that Lattices represented by matrices A, B are isomorphic (an isometric isomorphism exists) if and only represented by matrices if integer square matrices X, Y exist such that A = XBY, ...