update notations

content/CSE5313/CSE5313_L10.md

@@ -40,20 +40,20 @@ Let $G$ and $H$ be the generator and parity-check matrices of (any) linear code
 #### Lemma 1
 
 $$
-H G^T = 0
+H G^\top = 0
 $$
 
 <details>
 <summary>Proof</summary>
 
-By definition of generator matrix and parity-check matrix, $forall e_i\in H$, $e_iG^T=0$.
+By definition of generator matrix and parity-check matrix, $forall e_i\in H$, $e_iG^\top=0$.
 
-So $H G^T = 0$.
+So $H G^\top = 0$.
 </details>
 
 #### Lemma 2
 
-Any matrix $M\in \mathbb{F}_q^{(n-k)\times n}$ such that $\operatorname{rank}(M) = n - k$ and $M G^T = 0$ is a parity-check matrix for $C$ (i.e. $C = \ker M$).
+Any matrix $M\in \mathbb{F}_q^{(n-k)\times n}$ such that $\operatorname{rank}(M) = n - k$ and $M G^\top = 0$ is a parity-check matrix for $C$ (i.e. $C = \ker M$).
 
 <details>
 <summary>Proof</summary>
@@ -62,7 +62,7 @@ It is sufficient to show that the two statements
 
 1. $\forall c\in C, c=uG, u\in \mathbb{F}^k$
 
-$M c^T = M(uG)^T = M(G^T u^T) = 0$ since $M G^T = 0$.
+$M c^\top = M(uG)^\top = M(G^\top u^\top) = 0$ since $M G^\top = 0$.
 
 Thus $C \subseteq \ker M$.
 
@@ -84,15 +84,15 @@ We proceed by applying the lemma 2.
 
 1. $\operatorname{rank}(H) = n - k$ since $H$ is a Vandermonde matrix times a diagonal matrix with no zero entries, so $H$ is invertible.
 
-2. $H G^T = 0$.
+2. $H G^\top = 0$.
 
-note that $\forall$ row $i$ of $H$, $0\leq i\leq n-k-1$, $\forall$ column $j$ of $G^T$, $0\leq j\leq k-1$
+note that $\forall$ row $i$ of $H$, $0\leq i\leq n-k-1$, $\forall$ column $j$ of $G^\top$, $0\leq j\leq k-1$
 
 So
 
 $$
 \begin{aligned}
-H G^T &= \begin{bmatrix}
+H G^\top &= \begin{bmatrix}
 1 & 1 & \cdots & 1\\
 \alpha_1 & \alpha_2 & \cdots & \alpha_n\\
 \alpha_1^2 & \alpha_2^2 & \cdots & \alpha_n^2\\

content/CSE5313/CSE5313_L11.md

@@ -101,7 +101,7 @@ $$
 
 Let $\mathcal{C}=[n,k,d]_q$.
 
-The dual code of $\mathcal{C}$ is $\mathcal{C}^\perp=\{x\in \mathbb{F}^n_q|xc^T=0\text{ for all }c\in \mathcal{C}\}$.
+The dual code of $\mathcal{C}$ is $\mathcal{C}^\perp=\{x\in \mathbb{F}^n_q|xc^\top=0\text{ for all }c\in \mathcal{C}\}$.
 
 <details>
 <summary>Example</summary>
@@ -151,7 +151,7 @@ So $\langle f,h\rangle=0$.
 <details>
 <summary>Proof for the theorem</summary>
 
-Recall that the dual code of $\operatorname{RM}(r,m)^\perp=\{x\in \mathbb{F}_2^m|xc^T=0\text{ for all }c\in \operatorname{RM}(r,m)\}$.
+Recall that the dual code of $\operatorname{RM}(r,m)^\perp=\{x\in \mathbb{F}_2^m|xc^\top=0\text{ for all }c\in \operatorname{RM}(r,m)\}$.
 
 So $\operatorname{RM}(m-r-1,m)\subseteq \operatorname{RM}(r,m)^\perp$.
 

content/CSE5313/CSE5313_L14.md

@@ -230,7 +230,7 @@ Step 1: Arrange the $B=\binom{k+1}{2}+k(d-k)$ symbols in a matrix $M$ follows:
 $$
 M=\begin{pmatrix}
 S & T\\
-T^T & 0
+T^\top & 0
 \end{pmatrix}\in \mathbb{F}_q^{d\times d}
 $$
 
@@ -267,15 +267,15 @@ Repair from (any) nodes $H = \{h_1, \ldots, h_d\}$.
 
 Newcomer contacts each $h_j$: “My name is $i$, and I’m lost.”
 
-Node $h_j$ sends $c_{h_j}M c_i^T$ (inner product).
+Node $h_j$ sends $c_{h_j}M c_i^\top$ (inner product).
 
-Newcomer assembles $C_H Mc_i^T$.
+Newcomer assembles $C_H Mc_i^\top$.
 
 $CH$ invertible by construction!
 
-- Recover $Mc_i^T$.
+- Recover $Mc_i^\top$.
 
-- Recover $c_i^TM$ ($M$ is symmetric)
+- Recover $c_i^\topM$ ($M$ is symmetric)
 
 #### Reconstruction on Product-Matrix MBR codes
 
@@ -292,9 +292,9 @@ DC assembles $C_D M$.
 
 $\Psi_D$ invertible by construction.
 
-- DC computes $\Psi_D^{-1}C_DM = (S+\Psi_D^{-1}\Delta_D^T, T)$
+- DC computes $\Psi_D^{-1}C_DM = (S+\Psi_D^{-1}\Delta_D^\top, T)$
 - DC obtains $T$.
-- Subtracts $\Psi_D^{-1}\Delta_D T^T$ from $S+\Psi_D^{-1}\Delta_D T^T$ to obtain $S$.
+- Subtracts $\Psi_D^{-1}\Delta_D T^\top$ from $S+\Psi_D^{-1}\Delta_D T^\top$ to obtain $S$.
 
 <details>
 <summary>Fill an example here please.</summary>

content/CSE5313/CSE5313_L19.md

@@ -0,0 +1,232 @@
+# CSE5313 Coding and information theory for data science (Lecture 19)
+
+## Private information retrieval
+
+### Problem setup
+
+Premise:
+
+- Database $X = \{x_1, \ldots, x_m\}$, each $x_i \in \mathbb{F}_q^k$ is a "file" (e.g., medical record).
+- $X$ is coded $X \mapsto \{y_1, \ldots, y_n\}$, $y_j$ stored at server $j$.
+- The user (physician) wants $x_i$.
+- The user sends a query $q_j \sim Q_j$ to server $j$.
+- Server $j$ responds with $a_j \sim A_j$.
+
+Decodability:
+
+- The user can retrieve the file: $H(X_i | A_1, \ldots, A_n) = 0$.
+
+Privacy:
+
+- $i$ is seen as $i \sim U = U_{m}$, reflecting server's lack of knowledge.
+- $i$ must be kept private: $I(Q_j; U) = 0$ for all $j \in n$.
+
+> In short, we want to retrieve $x_i$ from the servers without revealing $i$ to the servers.
+
+### Private information retrieval from Replicated Databases
+
+#### Simple case, one server
+
+Say $n = 1, y_1 = X$.
+
+- All data is stored in one server.
+- Simple solution:
+- $q_1 =$ "send everything".
+- $a_1 = y_1 = X$.
+
+Theorem: Information Theoretic PIR with $n = 1$ can only be achieved by downloading the entire database.
+
+- Can we do better if $n > 1$?
+
+#### Collusion parameter
+
+Key question for $n > 1$: Can servers collude?
+
+- I.e., does server $j$ see any $Q_\ell$, $\ell \neq j$?
+- Key assumption:
+  - Privacy parameter $z$.
+  - At most $z$ servers can collude.
+  - $z = 1\implies$ No collusion.
+- Requirement for $z = 1$: $I(Q_j; U) = 0$ for all $j \in n$.
+- Requirement for a general $z$:
+  - $I(Q_\mathcal{T}; U) = 0$ for all $\mathcal{T} \in n$, $|\mathcal{T}| \leq z$, where $Q_\mathcal{T} = Q_\ell$ for all $\ell \in \mathcal{T}$.
+- Motivation:
+  - Interception of communication links.
+  - Data breaches.
+
+Other assumptions:
+
+- Computational Private information retrieval (even all the servers are hacked, still cannot get the information -> solve np-hard problem):
+- Non-zero MI
+
+#### Private information retrieval from 2-replicated databases
+
+First PIR protocol: Chor et al. FOCS ‘95.
+
+- The data $X = \{x_1, \ldots, x_m\}$ is replicated on two servers.
+  - $z = 1$, i.e., no collusion.
+- Protocol: User has $i \sim U_{m}$.
+  - User generates $r \sim U_{\mathbb{F}_q^m}$.
+  - $q_1 = r, q_2 = r + e_i$ ($e_i \in \mathbb{F}_q^m$ is the $i$-th unit vector, $q_2$ is equivalent to one-time pad encryption of $x_i$ with key $r$).
+  - $a_j = q_j X^\top = \sum_{\ell \in m} q_j, \ell x_\ell$
+  - Linear combination of the files according to the query vector $q_j$.
+- Decoding?
+  - $a_2 - a_1 = q_2 - q_1 X^\top = e_i X^\top = x_i$.
+- Download?
+  - $a_j =$ size of file $\implies$ downloading **twice** the size of the file.
+- Privacy?
+  - Since $z = 1$, need to show $I(U; Q_i) = 0$.
+    - $I(U; Q_1) = I(e_U; F) = 0$ since $U$ and $F$ are independent.
+    - $I(U; Q_2) = I(e_U; F + e_U) = 0$ since this is one-time pad!
+
+##### Parameters and notations in PIR
+
+Parameters of the system:
+
+- $n =$ # servers (as in storage).
+- $m =$ # files.
+- $k =$ size of each file (as in storage).
+- $z =$ max. collusion (as in secret sharing).
+- $t =$ # of answers required to obtain $x_i$ (as in secret sharing).
+  - $n - t$ servers are “stragglers”, i.e., might not respond.
+
+Figures of merit:
+
+- PIR-rate = $\#$ desired symbols / $\#$ downloaded symbols
+- PIR-capacity = largest possible rate.
+
+Notaional conventions:
+
+-The dataset $X = \{x_j\}_{j \in m} = \{x_{j, \ell}\}_{(j, \ell) \in [m] \times [k]}$ is seen as a vector in $\mathbb{F}_q^{mk}$.
+
+- Index $\mathbb{F}_q^{mk}$ using $[m] \times [k]$, i.e., $x_{j, \ell}$ is the $\ell$-th symbol of the $j$-th file.
+
+#### Private information retrieval from 4-replicated databases
+
+Consider $n = 4$ replicated servers, file size $k = 2$, collusion $z = 1$.
+
+Protocol: User has $i \sim U_{m}$.
+
+- Fix distinct nonzero $\alpha_1, \ldots, \alpha_4 \in \mathbb{F}_q$.
+- Choose $r \sim U_{\mathbb{F}_q^{2m}}$.
+- User sends $q_j = e_{i, 1} + \alpha_j e_{i, 2} + \alpha_j^2 r$ to each server $j$.
+- Server $j$ responds with
+  $$
+  a_j = q_j X^\top = e_{i, 1} X^\top + \alpha_j e_{i, 2} X^\top + \alpha_j^2 r X^\top
+  $$
+  - This is an evaluation at $\alpha_j$ of the polynomial $f_i(w) = x_{i, 1} + x_{i, 2} \cdot w + r \cdot w^2$.
+  - Where $r$ is some random combination of the entries of $X$.
+- Decoding?
+  - Any 3 responses suffice to interpolate $f_i$ and obtain $x_i = x_{i, 1}, x_{i, 2}$.
+  - $\implies t = 3$, (one straggler is allowed)
+- Privacy?
+  - Does $q_j = e_{i, 1} + \alpha_j e_{i, 2} + \alpha_j^2 r$ look familiar?
+  - This is a share in [ramp scheme](CSE5313_L18.md#scheme-2-ramp-secret-sharing-scheme-mceliece-sarwate-scheme) with vector messages $m_1 = e_{i, 1}, m_2 = e_{i, 2}, m_i \in \mathbb{F}_q^{2m}$.
+  - This is equivalent to $2m$ "parallel" ramp scheme over $\mathbb{F}_q$.
+  - Each one reveals nothing to any $z = 1$ shareholders $\implies$ Private!
+
+### Private information retrieval from general replicated databases
+
+$n$ servers, $m$ files, file size $k$, $X \in \mathbb{F}_q^{mk}$.
+
+Server decodes $x_i$ from any $t$ responses.
+
+Any $\leq z$ servers might collude to infer $i$ ($z < t$).
+
+Protocol: User has $i \sim U_{m}$.
+
+- User chooses $r_1, \ldots, r_z \sim U_{\mathbb{F}_q^{mk}}$.
+- User sends $q_j = \sum_{\ell=1}^k e_{i, \ell} \alpha_j^{\ell-1} + \sum_{\ell=1}^z r_\ell \alpha_j^{k+\ell-1}$ to each server $j$.
+- Server $j$ responds with $a_j = q_j X^\top = f_i(\alpha_j)$.
+  - $f_i(w) = \sum_{\ell=1}^k e_{i, \ell} X^\top w^{\ell-1} + \sum_{\ell=1}^z r_\ell X^\top w^{k+\ell-1}$ (random combinations of $X$).
+  - Caveat: must have $t = k + z$.
+  - $\implies \deg f_i = k + z - 1 = t - 1$.
+- Decoding?
+  - Interpolation from any $t$ evaluations of $f_i$.
+- Privacy?
+  - Against any $z = t - k$ colluding servers, immediate from the proof of the ramp scheme.
+
+PIR-rate?
+
+- Each $a_j$ is a single field element.
+- Download $t = k + z$ elements in $\mathbb{F}_q$ in order to obtain $x_i \in \mathbb{F}_q^k$.
+- $\implies$ PIR-rate = $\frac{k}{k+z} = \frac{k}{t}$.
+
+#### Theorem: PIR-capacity for general replicated databases
+
+The PIR-capacity for $n$ replicated databases with $z$ colluding servers, $n - t$ unresponsive servers, and $m$ files is $C = \frac{1-\frac{z}{t}}{1-(\frac{z}{t})^m}$.
+
+- When $m \to \infty$, $C \to 1 - \frac{z}{t} = \frac{t-z}{t} = \frac{k}{t}$.
+- The above scheme achieves PIR-capacity as $m \to \infty$
+
+### Private information retrieval from coded databases
+
+#### Problem setup:
+
+Example:
+
+- $n = 3$ servers, $m$ files $x_j$, $x_j = x_{j, 1}, x_{j, 2}$, $k = 2$, and $q = 2$.
+- Code each file with a parity code: $x_{j, 1}, x_{j, 2} \mapsto x_{j, 1}, x_{j, 2}, x_{j, 1} + x_{j, 2}$.
+- Server $j \in 3$ stores all $j$-th symbols of all coded files.
+
+Queries, answers, decoding, and privacy must be tailored for the code at hand.
+
+With respect to a code $C$ and parameters $n, k, t, z$, such scheme is called coded-PIR.
+
+- The content for server $j$ is denoted by $c_j = c_{j, 1}, \ldots, c_{j, m}$.
+- $C$ is usually an MDS code.
+
+#### Private information retrieval from parity-check codes
+
+Example:
+
+ Say $z = 1$ (no collusion).
+
+- Protocol: User has $i \sim U_{m}$.
+- User chooses $r_1, r_2 \sim U_{\mathbb{F}_2^m}$.
+- Two queries to each server:
+  - $q_{1, 1} = r_1 + e_i$, $q_{1, 2} = r_2$.
+  - $q_{2, 1} = r_1$, $q_{2, 2} = r_2 + e_i$.
+  - $q_{3, 1} = r_1$, $q_{3, 2} = r_2$.
+- Server $j$ responds with $q_{j, 1} c_j^\top$ and $q_{j, 2} c_j^\top$.
+- Decoding?
+  - $q_{1, 1} c_1^\top + q_{2, 1} c_2^\top + q_{3, 1} c_3^\top = r_1 c_1 + c_2 + c_3 + e_i c_1^\top = r_1 \cdot 0^\top + x_{i, 1} = x_{i, 1}$.
+  - $q_{1, 1} c_1^\top + q_{2, 1} c_2^\top + q_{3, 1} c_3^\top = r_1 \cdot 0^\top + x_{i, 1} = x_{i, 1}$.
+  - $q_{1, 2} c_1^\top + q_{2, 2} c_2^\top + q_{3, 2} c_3^\top = r_2 c_1 + c_2 + c_3^\top + e_i c_2^\top = x_{i, 2}$.
+- Privacy?
+  - Every server sees two uniformly random vectors in $\mathbb{F}_2^m$.
+
+<details>
+<summary>Proof from coding-theoretic interpretation</summary>
+
+Let $G = g_1^\top, g_2^\top, g_3^\top$ be the generator matrix. 
+
+- For every file $x_j = x_{j, 1}, x_{j, 2}$ we encode $x_j G = (x_{j, 1} g_1^\top, x_{j, 2} g_2^\top, x_{j, 1} g_3^\top) = (c_{j, 1}, c_{j, 2}, c_{j, 3})$.
+- Server $j$ stores $X g_j^\top = (x_1^\top, \ldots, x_m^\top)^\top g_j^\top = (c_{j, 1}, \ldots, c_{j, m})^\top$.
+
+- By multiplying by $r_1$, the servers together store a codeword in $C$:
+  - $r_1 X g_1^\top, r_1 X g_2^\top, r_1 X g_3^\top = r_1 X G$.
+- By replacing one of the $r_1$’s by $r_1 + e_i$, we introduce an error in that entry:
+  - $\left((r_1 + e_i) X g_1^\top, r_1 X g_2^\top, r_1 X g_3^\top\right) = r_1 X G + (e_i X g_1^\top, 0,0)$.
+- Downloading this “erroneous” word from the servers and multiply by $H = h_1^\top, h_2^\top, h_3^\top$ be the parity-check matrix.
+
+$$
+\begin{aligned}
+\left((r_1 + e_i) X g_1^\top, r_1 X g_2^\top, r_1 X g_3^\top\right) H^\top &= \left(r_1 X G + (e_i X g_1^\top, 0,0)\right) H^\top \\
+&= r_1 X G H^\top + (e_i X g_1^\top, 0,0) H^\top \\
+&= 0 + x_{i, 1} g_1^\top \\
+&= x_{i, 1}.
+\end{aligned}
+$$
+
+> In homework we will show tha this work with any MDS code ($z=1$).
+
+- Say we obtained $x_{i, 1} g_1^\top, \ldots, x_{i, k} g_k^\top$ (𝑑 − 1 at a time, how?).
+- $x_{i, 1} g_1^\top, \ldots, x_{i, k} g_k^\top = x_{i, B}$, where $B$ is a $k \times k$ submatrix of $G$.
+- $B$ is a $k \times k$ submatrix of $G$ $\implies$ invertible! $\implies$ Obtain $x_{i}$.
+
+</details>
+
+> [!TIP]
+>
+> error + known location $\implies$ erasure. $d = 2 \implies$ 1 erasure is correctable.

content/CSE5313/CSE5313_L6.md

@@ -92,10 +92,10 @@ Two equivalent ways to constructing a linear code:
 
 - A **parity check** matrix $H\in \mathbb{F}^{(n-k)\times n}$ with $(n-k)$ rows and $n$ columns.
   $$
-  \mathcal{C}=\{c\in \mathbb{F}^n:Hc^T=0\}
+  \mathcal{C}=\{c\in \mathbb{F}^n:Hc^\top=0\}
   $$
   - The right kernel of $H$ is $\mathcal{C}$.
-  - Multiplying $c^T$ by $H$ "checks" if $c\in \mathcal{C}$.
+  - Multiplying $c^\top$ by $H$ "checks" if $c\in \mathcal{C}$.
 
 ### Encoding of linear codes
 
@@ -144,7 +144,7 @@ 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.
 
 $$
-H(y+e)^T=H(y+e)=Hx+He=He
+H(y+e)^\top=H(y+e)=Hx+He=He
 $$
 
 ### Syndrome decoding
@@ -215,7 +215,7 @@ Fourth row is $\mathcal{C}+(00100)$.
 
 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 $H(y_1'+e)^T=H(y_2'+e)^T$.
+Same syndrome if $H(y_1'+e)^\top=H(y_2'+e)^\top$.
 
 Entries in different rows have different syndrome.
 

content/CSE5313/CSE5313_L7.md

@@ -7,7 +7,7 @@ Let $\mathcal{C}= [n,k,d]_{\mathbb{F}}$ be a linear code.
 There are two equivalent ways to describe a linear code:
 
 1. A generator matrix $G\in \mathbb{F}^{k\times n}_q$ with $k$ rows and $n$ columns, entry taken from $\mathbb{F}_q$. $\mathcal{C}=\{xG|x\in \mathbb{F}^k\}$
-2. A parity check matrix $H\in \mathbb{F}^{(n-k)\times n}_q$ with $(n-k)$ rows and $n$ columns, entry taken from $\mathbb{F}_q$. $\mathcal{C}=\{c\in \mathbb{F}^n:Hc^T=0\}$
+2. A parity check matrix $H\in \mathbb{F}^{(n-k)\times n}_q$ with $(n-k)$ rows and $n$ columns, entry taken from $\mathbb{F}_q$. $\mathcal{C}=\{c\in \mathbb{F}^n:Hc^\top=0\}$
 
 ### Dual code
 
@@ -21,7 +21,7 @@ $$
 
 Also, the alternative definition is:
 
-1. $C^{\perp}=\{x\in \mathbb{F}^n:Gx^T=0\}$ (only need to check basis of $C$)
+1. $C^{\perp}=\{x\in \mathbb{F}^n:Gx^\top=0\}$ (only need to check basis of $C$)
 2. $C^{\perp}=\{xH|x\in \mathbb{F}^{n-k}\}$
 
 By rank-nullity theorem, $dim(C^{\perp})=n-dim(C)=n-k$.
@@ -87,7 +87,7 @@ Assume minimum distance is $d$. Show that every $d-1$ columns of $H$ are indepen
 
 - Fact: In linear codes minimum distance is the minimum weight ($d_H(x,y)=w_H(x-y)$).
 
-Indeed, if there exists a $d-1$ columns of $H$ that are linearly dependent, then we have $Hc^T=0$ for some $c\in \mathcal{C}$ with $w_H(c)<d$.
+Indeed, if there exists a $d-1$ columns of $H$ that are linearly dependent, then we have $Hc^\top=0$ for some $c\in \mathcal{C}$ with $w_H(c)<d$.
 
 Reverse are similar.
 
@@ -130,7 +130,7 @@ $k=2^m-m-1$.
 
 Define the code by encoding function:
 
-$E(x): \mathbb{F}_2^m\to \mathbb{F}_2^{2^m}=(xy_1^T,\cdots,xy_{2^m}^T)$ ($y\in \mathbb{F}_2^m$)
+$E(x): \mathbb{F}_2^m\to \mathbb{F}_2^{2^m}=(xy_1^\top,\cdots,xy_{2^m}^\top)$ ($y\in \mathbb{F}_2^m$)
 
 Space of codewords is image of $E$.
 

content/CSE5313/CSE5313_L8.md

@@ -258,7 +258,7 @@ Algorithm:
 - Begin with $(n-k)\times (n-k)$ identity matrix.
 - Assume we choose columns $h_1,h_2,\ldots,h_\ell$ (each $h_i$ is in $\mathbb{F}^n_q$)
 - Then next column $h_{\ell}$ must not be in the space of any previous $d-2$ columns.
-  - $h_{\ell}$ cannot be written as $[h_1,h_2,\ldots,h_{\ell-1}]x^T$ for $x$ of Hamming weight at most $d-2$.
+  - $h_{\ell}$ cannot be written as $[h_1,h_2,\ldots,h_{\ell-1}]x^\top$ for $x$ of Hamming weight at most $d-2$.
   - So the ineligible candidates for $h_{\ell}$ is:
     - $B_{\ell-1}(0,d-2)=\{x\in \mathbb{F}^{\ell-1}_q: d_H(0,x)\leq d-2\}$.
     - $|B_{\ell-1}(0,d-2)|=\sum_{i=0}^{d-2}\binom{\ell-1}{i}(q-1)^i$, denoted by $V_q(\ell-1, d-2)$.

content/CSE5313/CSE5313_L9.md

@@ -148,15 +148,15 @@ The generator matrix for Reed-Solomon code is a Vandermonde matrix $V(a_1,a_2,\l
 
 Fact: $V(a_1,a_2,\ldots,a_n)$ is invertible if and only if $a_1,a_2,\ldots,a_n$ are distinct. (that's how we choose $a_1,a_2,\ldots,a_n$)
 
-The parity check matrix for Reed-Solomon code is also a Vandermonde matrix $V(a_1,a_2,\ldots,a_n)^T$ with scalar multiples of the columns.
+The parity check matrix for Reed-Solomon code is also a Vandermonde matrix $V(a_1,a_2,\ldots,a_n)^\top$ with scalar multiples of the columns.
 
 Some technical lemmas:
 
 Let $G$ and $H$ be the generator and parity-check matrices of (any) linear code
 $C = [n, k, d]_{\mathbb{F}_q}$. Then:
 
-I. Then $H G^T = 0$.
-II. Any matrix $M \in \mathbb{F}_q^{n-k \times k}$ such that $\rank(M) = n - k$ and $M G^T = 0$ is a parity-check matrix for $C$ (i.e. $C = \ker M$).
+I. Then $H G^\top = 0$.
+II. Any matrix $M \in \mathbb{F}_q^{n-k \times k}$ such that $\rank(M) = n - k$ and $M G^\top = 0$ is a parity-check matrix for $C$ (i.e. $C = \ker M$).
 
 ## Reed-Muller code
 

content/CSE5313/_meta.js

@@ -22,4 +22,5 @@ export default {
     CSE5313_L16: "CSE5313 Coding and information theory for data science (Exam Review)",
     CSE5313_L17: "CSE5313 Coding and information theory for data science (Lecture 17)",
     CSE5313_L18: "CSE5313 Coding and information theory for data science (Lecture 18)",
+    CSE5313_L19: "CSE5313 Coding and information theory for data science (Exam Review)",
 }