update notations

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 $E$.