updates

content/CSE5313/CSE5313_L2.md

@@ -267,11 +267,9 @@ Hamming distance is a metric.
 
 ### Level of error handling
 
-error detection
-
-erasure correction
-
-error correction
+- error detection
+- erasure correction
+- error correction
 
 Erasure: replacement of an entry by $*\not\in F$.
 
@@ -283,11 +281,27 @@ Example: If $d_H(\mathcal{C})=d$.
 
 Theorem: If $d_H(\mathcal{C})=d$, then there exists $f:F^n\to \mathcal{C}\cap \{\text{"error detected"}\}$. that detects every patter of $\leq d-1$ errors correctly.
 
-\* track lost *\
+- That is, we can identify if the channel introduced at most $d-1$ errors.
+- No decoding is needed.
 
 Idea:
 
-Since $d_H(\mathcal{C})=d$, one needs $\geq d$ errors to cause "confusion$.
+Since $d_H(\mathcal{C})=d$, one needs $\geq d$ errors to cause "confusion".
+
+<details>
+<summary>Proof</summary>
+
+The function
+$$
+f(y)=\begin{cases}
+y\text{  if }y\in \mathcal{C}\\
+\text{"error detected"} & \text{otherwise}
+\end{cases}
+$$
+
+will only fails if there are $\geq d$ errors.
+
+</details>
 
 #### Erasure correction
 
@@ -295,7 +309,11 @@ Theorem: If $d_H(\mathcal{C})=d$, then there exists $f:\{F^n\cup \{*\}\}\to \mat
 
 Idea:
 
-\* track lost *\
+Suppose $d=4$.
+
+If $4$ erasures occurred, there might be two possible codewords $c,c'\in \mathcal{C}$.
+
+If $\leq 3$ erasures occurred, there is only one possible codeword $c\in \mathcal{C}$.
 
 #### Error correction