update notations

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.