Update CSE5313_L26.md

content/CSE5313/CSE5313_L26.md

@@ -33,7 +33,7 @@ mRNA carries info to Ribosome as codons of length 3 over GUCA.
 
 2nd Chargaff rule:
 
-- $#𝐴 ≈ #𝑇$ and $#G \approx #C$ in each strands.
+- $#A \approx #T$ and $#G \approx #C$ in each strands.
 - Can be explained via tandem duplications.
   - $GCAGCATT \implies GCAGCAGCATT$.
   - Occur naturally during cell mitosis.
@@ -369,30 +369,31 @@ This implies that $n-\log |\mathcal{C}|\geq n-\log(n+1)-\log|\mathcal{C}_j|\geq
 
 Corollary: In the relevant regime $t=O(n^{1-\epsilon})$, we have $\Omega(t\log n)$ redundancy.
 
-TRACK LOST HERE
+### t-break codes: Main ideas.
 
-𝑡-break codes: Main ideas.
-• Encoding:
-– Need multiple markers across the codeword.
-– Construct an adjacency matrix 𝐴 of markers to record their order.
-– Append 𝑅𝑆2𝑡 𝐴 to the codeword (as in the sliced channel).
-• Decoding (from 𝑡 + 1 fragments):
-– Locate all surviving markers, and locate 𝑅𝑆2𝑡 𝐴 ′.
-– Build an approximate adjacency matrix 𝐴
-′
-from surviving markers (𝑑𝐻 𝐴, 𝐴′ ≤ 2𝑡).
-– Correct 𝐴
-′
-, 𝑅𝑆2𝑡 𝐴 ′ ↦ 𝐴 , 𝑅𝑆2𝑡 𝐴 .
-– Order the fragments correctly using 𝐴.
-• Tools:
-– Random encoding (to have many markers).
-– Mutually uncorrelated codes (so that markers will not overlap).
+Encoding:
 
-Tool: Mutually uncorrelated codes.
-• Want: Markers not to overlap.
-• Solution: Take markers from a Mutually Uncorrelated Codes (existing notion).
-– A code ℳ is called mutually uncorrelated if no suffix of any 𝑚𝑖 ∈ ℳ if a prefix of another
+- Need multiple markers across the codeword.
+- Construct an adjacency matrix 𝐴 of markers to record their order.
+- Append $RS_{2t}(A)$ to the codeword (as in the sliced channel).
+
+Decoding (from $t + 1$ fragments):
+
+- Locate all surviving markers, and locate $RS_{2t}(A)'$.
+- Build an approximate adjacency matrix $A'$ from surviving markers $(d_H(A, A' )\leq  2t)$.
+- Correct $(A',RS_{2t}(A)')\mapsto (A,RS_{2t}(A))$.
+- Order the fragments correctly using $A$.
+
+Tools:
+
+- Random encoding (to have many markers).
+- Mutually uncorrelated codes (so that markers will not overlap).
+
+#### Tool: Mutually uncorrelated codes.
+
+- Want: Markers not to overlap.
+- Solution: Take markers from a Mutually Uncorrelated Codes (existing notion).
+  - A code $\mathcal{M}$ is called mutually uncorrelated if no suffix of any 𝑚𝑖 ∈ ℳ if a prefix of another
 𝑚𝑗 ∈ ℳ (including 𝑖 = 𝑗).
 – Many constructions exist.
 • Theorem: For any integer ℓ there exists a mutually uncorrelated code 𝐶𝑀𝑈 of length