# Document 10572790

```Error Correcting Codes: Combinatorics, Algorithms and Applications
(Fall 2007)
Lecture 42: Achieving List-Decoding Capacity
December 7, 2007
Lecturer: Atri Rudra
Scribe: Sandipan Kundu
1 Introduction
!&quot;\$# as % &amp;(')*+
.
factoring out (E(X))
mod (E(X))
is poly over - is such
.0/1 and , 32.4/1 for at least
- , 27&amp;+98 # .
If ,
that ,
values of 6 , which implies ,
5
;:=&lt;&gt; over ? &lt; A@B .
Find the roots of
-+&lt; &quot; , 27&amp;DCFEHG % &amp; ,
Lemma 1.1. There exists a irreducible polynomial E(X) such that ,
, -AS.4&lt; /1&quot; , 3:H2I/ &amp;JCFEHG 32% .4/1&amp; KLM&gt;/ .4/A;:H/AKM&gt;/N &quot;O#QPR S.D/T S.DS.4/T+/1:UK/T KM&gt;/12I .4&quot;\$/1 #
&quot;O# which in turn implies
&quot; and ,
&quot; , then it implies
,
,
If ,
V &amp;X&quot;W , &amp;K , 27&amp;+ has Y[ZU5 roots.
G]\&gt;^ V _`+aUbIKbcAE , d_fe as 5gih j . (Done)
We know
, &amp; , 27&amp;+ &quot;\$# which implies , &amp; , 32I&amp;; &quot;O# .
In other words,
&amp;
&amp;l&quot;W +m&lt;( n &lt; J@oB for general C we have, pS, 32IqlrNK , 2sqlrut0v&gt;(w(w&gt;w&gt; , 2sqlrut7q!xyvK+z .
If ,
, 327qlisrNaK root
, 2sqlofrut0k v&gt;(w(w&gt;w&gt; , 2sqlrut7q!xyvKover
+{8 #
&lt;xyv
@
B
S
&gt;
(
w
&gt;
w
&gt;
w
_fe~} q!xyv of degree }fa .
&lt;
T
&lt;
|
Find roots of
over ? &lt; maximum number of roots
-+&lt; &quot; , 27&amp;DCFEHG % &amp; ,
Lemma 1.2. There exists a irreducible polynomial E(X) such that ,
k S needs to be non-zero or [ &lt; does’t divide .
_ degree of Y in+
(1)
degree of Y in
_ eV } by choice of D
(2)
+a
In fact for constant rate code, h  6T
&amp; &quot; &lt;xyv &amp;2 is irreducible over ? &lt; .
Lemma 1.3. %
 .
Exact characterization of irreducible polynomial of the form
&amp;A&lt;8 , for , &amp; ? &lt; 
Lemma 1.4. ,
SK &lt; &quot; o &lt; K &lt;
Proof. Hint: Show
Proof. Proof of lemma (1.1) using (1.4)
Done if
&amp;&lt;xyv{2
True if %
27&amp;8 # CFEUG % &amp;
, &amp; &lt; &lt;  , 27
8 # CFEHG % &amp;
P
P % &amp;AG 6No6 G]\   &lt;Kxyv &amp;2y
1
(3)
(4)
(5)
In for number of folds,
C
general,

5Xg bc /   v +aO&iexcl;j +
2
```