# 定義

### Lattice

Given $n$ linearly independent vectors $\mathbf{b_1}, \mathbf{b_2}, \cdots, \mathbf{b_n} \in \mathbb{R^m}$, the lattice generated by them is defined as

L(\mathbf{b_1}, \cdots, \mathbf{b_n}) \overset{\text{def}}{=} \Big{\{}\sum_{i=1}^{n}x_i\mathbf{b_i} \ | \ x_i \in \mathbb{Z}\Big{\}}

We can write $\mathbf{b_1}, \cdots, \mathbf{b_n}$ as a matrix $\mathbf{B}$ for convenience

\mathbf{B} = \begin{pmatrix} | & & | \\ \mathbf{b_1} & \cdots & \mathbf{b_n} \\ | & & | \end{pmatrix}
L(\mathbf{B}) \overset{\text{def}}{=} \{\mathbf{Bx} \ | \ \mathbf{x} \in \mathbb{Z}^n \}

We call $\{\mathbf{b_1}, \cdots, \mathbf{b_n}\}$ a basis of lattice
We call $n$ the rank of lattice
We call $m$ the dimension of lattice
When $n = m$, we call the lattice a full-rank lattice

Lattice 跟 Vector Space 很像，但 Lattice 是 vectors 整數倍的 linear combination

### Unimodular Matrix

A matrix $\mathbf{U} \in \mathbb{Z}^{n \times n}$ is unimodular if $|\text{det}(\mathbf{U})| = 1$

#### 性質 ( property )

1. $\mathbf{U}$ is unimodular $\Leftrightarrow$ $\mathbf{U}^{-1}$ is unimodular

### Fundamental Parallelepiped

Given $n$ linearly independent vectors $\mathbf{b_1}, \mathbf{b_2}, \cdots, \mathbf{b_n} \in \mathbb{R^m}$, their fundamental parallelepiped is defined as

P(\mathbf{b_1}, \cdots, \mathbf{b_n}) \overset{\text{def}}{=} \Big{\{}\sum_{i=1}^{n}x_i\mathbf{b_i} \ | \ x_i \in \mathbb{R}, 0 \le x_i < 1\Big{\}}

### Determinant of Lattice

Let $\mathbf{B}$ be the basis matrix of $L$

$\text{det}(L) = \text{det}(\mathbf{B}) = \text{vol}(P(\mathbf{B}))$

$\text{det}(L)$ 是個定值，所有的 bases 都有一樣的 determinant

### Minimum Distance of Lattice

$\lambda_1(L) = \underset{v \ \in \ L \setminus 0}{min} \|v\|$