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+# CSE5313 Coding and information theory for data science (Lecture 6)
+
+## Recap
+
+### Vector spaces and subspaces over finite fields
+
+$\mathbb{F}^n$ is a vector space over $\mathbb{F}$.
+
+With point-wise vector addition and scalar multiplication.
+
+<details>
+<summary>Example</summary>
+
+$\mathbb{F}_2^4$ is a vector space over $\mathbb{F}_2$.
+
+Let $v=\begin{pmatrix}
+1 & 1 & 1 & 1
+\end{pmatrix}$
+
+Then $v$ is a vector in $\mathbb{F}_2^4$ that's "orthogonal" to itself.
+
+$v\cdot v=1+1+1+1=4=0$ in $\mathbb{F}_2$.
+
+In general field, the dual space and space may intersect non-trivially.
+
+</details>
+
+Let $V$ be a subspace of $\mathbb{F}^n$.
+
+$V$ is a subgroup of $\mathbb{F}^n$ under vector addition $(\mathbb{F}^n,+)$.
+
+- Apply the theorem: If $H$ is finite, non-empty, and closed under the operation of $G$, then $H$ is a subgroup of $G$.
+
+<details>
+<summary>Proof</summary>
+Since $H\subseteq G$, $H$ is non-empty and closed under the operation of $G$ and finite, then $H\leq G$.
+
+left to show:
+
+Associativity: inherited from $G$.
+
+Unit element: $0\in H$.
+
+Consider $a\in H$, $a,a^2,a^3,\cdots$ are in $H$. Since $H$ is finite, there exists $i,j\in\mathbb{N}$ such that $a^i=a^j$.
+
+Then $a^i=a^j\iff a^{i-j}=e\in H$.
+
+Inverses: $a^{-1}\in H$.
+
+Automatically holds for unit element traversing.
+
+</details>
+
+> Is every subgroup of $\mathbb{F}^n$ a subspace?
+
+<details>
+<summary>Answer</summary>
+No.
+
+Consider $F_4=\{0,1,x,x+1\}$ (field extension of $\mathbb{F}_2$ with $p(x)=x$).
+
+$F_4^2=\{(a,b):a,b\in F_4\}$, $\{(0,0),(1,1)\}$ is a subgroup of $(F_4^2,+)$.
+
+But the span of $F_4\{(1,1)\}$ is $\{(0,0),(1,1),(x,x),(x+1,x+1)\}\neq \{(0,0),(1,1)\}$, which is not a subspace of $F_4^2$.
+
+</details>
+
+Cosets in this definition are called Affine subspaces.
+
+$$
+V+a=\{v+a:v\in V\}\text{ for some }a\in \mathbb{F}^n
+$$
+
+## New content
+
+### Linear codes
+
+A linear code $\mathcal{C}$ is a subspace of $\mathbb{F}^n$ over $\mathbb{F}$.
+
+- The dimension of $\mathcal{C}$ is denoted by $k$.
+- The minimum Hamming distance of $\mathcal{C}$ is denoted by $d$.
+- Notation $\mathcal{C}= [n,k,d]_{\mathbb{F}}$.
+
+Two equivalent ways to constructing a linear code:
+
+- A **generator matrix** $G\in \mathbb{F}^{k\times n}$ with $k$ rows and $n$ columns.
+  $$
+  \mathcal{C}=\{xG:x\in \mathbb{F}^k\}
+  $$
+  - The left image of $G$ is $\mathcal{C}$.
+  - The rows of $G$ are a basis for $\mathcal{C}$.
+
+- A **parity check** matrix $H\in \mathbb{F}^{(n-k)\times n}$ with $(n-k)$ rows and $n$ columns.
+  $$
+  \mathcal{C}=\{c\in \mathbb{F}^n:Hc^T=0\}
+  $$
+  - The right kernel of $H$ is $\mathcal{C}$.
+  - Multiplying $c^T$ by $H$ "checks" if $c\in \mathcal{C}$.
+
+### Encoding of linear codes
+
+Reminder:
+
+- Encoding is the process of mapping a message $u\in \mathbb{F}^k$ to a codeword $c\in \mathcal{C}\subseteq \mathbb{F}^n$.
+
+$E: \mathbb{F}^k\to \mathcal{C}$ is a linear map.
+
+Let $\mathcal{C}= [n,k,d]_{\mathbb{F}}$ be a linear code with generator matrix $G\in \mathbb{F}^{k\times n}$.
+
+- Encoding is given by $E(x)=xG$.
+- It is injective (1-1). Suppose otherwise, then there exists $x_1,x_2\in \mathbb{F}^k$ such that $x_1G=x_2G$. Then $x_1G-x_2G=0\implies (x_1-x_2)G=0\implies x_1-x_2=0\implies x_1=x_2$. Therefore, $E$ is injective.
+
+So linear codes implies linear encoding: $E(x)+E(y)=E(x+y)$.
+
+### Systematic codes
+
+Fact: Every $G\in \mathbb{F}^{k\times n}$ can be brought to the form $G_{sys}=(I|A)$ by
+
+- Row operations.
+- Permutation of columns.
+
+Fact $\{xG|x\in \mathbb{F}^k\}$ and $\{xG_{sys}|x\in \mathbb{F}^k\}$ are equivalent.
+
+- Same length $n$.
+- Same dimension $k$.
+- Same minimum Hamming distance $d$.
+
+Encoding a systematic code:
+
+- The input is a part of the output.
+- Efficient encoding
+- Immediate decoding (if no errors).
+
+### Codes, cosets, encoding, decoding
+
+Linear code $[n,k,d]_{\mathbb{F}}$ is a $k$ dimensional subspace of $\mathbb{F}^n$.
+
+Size of the code is $|\mathbb{F}|^k$.
+
+Encoding: $x\to xG$.
+
+Decoding: $(y+e)\to x$, $y=xG$.
+
+Use **syndrome** to identify which coset $\mathcal{C}_i$ that the noisy-code to $\mathcal{C}_i+e$ belongs to.
+
+$$
+H(y+e)^T=H(y+e)=Hx+He=He
+$$
+
+### Syndrome decoding
+
+- Heavily depends onn the linear structure of the code.
+
+Linear code $\mathcal{C}= [n,k,d]_{\mathbb{F}}$ is a $k$-dimensional subspace of $(\mathbb{F}^n,+)$.
+
+**Shift of** Linear code $[n,k,d]_{\mathbb{F}}$ is a $k$-dimensional affine subspace of $\mathbb{F}^n$.
+
+All cosets of the same size
+
+If $w_H(e)\leq \lfloor \frac{d-1}{2}\rfloor$, then it is possible to extract $y$ from $y+e$.
+
+by syndrome decoding, we can do better than exhaustive search.
+
+Idea:
+
+Let $y+e$ belogns to the coset $\mathcal{C}+e$.
+
+Moreover,$y_1+e$ and $y_2+e$ are in the same coset.
+
+#### Standard Array
+
+Let $\mathcal{C}= [n,k,d]_{\mathbb{F}}$ and denote $|F|=q$.
+
+- Then $|\mathcal{C}|=q^k$.
+- The number of cosets is $q^{n-k}$.
+
+Then we arrange all $q^n$ elements of $\mathbb{F}^n$ into a $q^{n-k}\times q^k$ array.
+
+- So that every row is a coset (including $\mathcal{C}$ itself)
+- Lightest word in each cosets on the leftmost column
+
+<details>
+<summary>Example</summary>
+
+Let $\mathbb{F}=\mathbb{Z}_2$ and $C=\{xG|x\in \mathbb{F}_2\}$
+
+$$
+G=\begin{pmatrix}
+1 & 0 & 1 & 1 & 0\\
+0 & 1 & 1 & 0 & 1
+\end{pmatrix}
+$$
+
+So $\mathcal{C}=\{00000,10110,01011,11101\}$.
+
+Then $G=[5,2,3]_2$.
+
+The standard array is:
+
+First row is $\mathcal{C}$.
+
+Second row is $\mathcal{C}+(00001)$,
+
+Third row is $\mathcal{C}+(00010)$.
+
+Fourth row is $\mathcal{C}+(00100)$.
+
+|00000|10110|01011|11101|
+|---|---|---|---|
+|00001|10111|01010|11100|
+|00010|10100|01001|11110|
+|00100|10010|01101|11000|
+
+</details>
+
+Any two elements in a row are of the form $y_1'=y_1+e$ and $y_2'=y_2+e$ for some $e\in \mathbb{F}^n$.
+
+Same syndrome if $H(y_1'+e)^T=H(y_2'+e)^T$.
+
+Entries in different rows have different syndrome.
+
+<details>
+<summary>Proof</summary>
+
+
+</details>
+
+Choose the lightest word in each coset on the leftmost column.
+
+Time complexity: $O(n(n-k))$. Space complexity: $n|F|^n$ space.
+
+Compare with exhaustive search: Time: $O(|F|^n)$.
+
+#### Syndrome decoding - Intuition
+
+Given 𝒚′, we identify the set 𝐶 + 𝒆 to which 𝒚′ belongs by computing the syndrome.
+• We identify 𝒆 as the coset leader (leftmost entry) of the row 𝐶 + 𝒆.
+• We output the codeword in 𝐶 which is closest (𝒄3) by subtracting 𝒆 from 𝒚′.
+
+#### Syndrome decoding - Formal
+
+Given $y'\in \mathbb{F}^n$, we identify the set $\mathcal{C}+e$ to which $y'$ belongs by computing the syndrome.
+
+We identify $e$ as the coset leader (leftmost entry) of the row $\mathcal{C}+e$.
+
+We output the codeword in $\mathcal{C}$ which is closest ($c_3$) by subtracting $e$ from $y'$.