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
!"$# as % &(')*+
.
factoring out (E(X))
mod (E(X))
is poly over - is such
.0/1 and , 32.4/1 for at least
- , 27&+98 # .
If ,
that ,
values of 6 , which implies ,
5
;:=<> over ? < A@B .
Find the roots of
-+< " , 27&DCFEHG % & ,
Lemma 1.1. There exists a irreducible polynomial E(X) such that ,
, -AS.4< /1" , 3:H2I/ &JCFEHG 32% .4/1& KLM>/ .4/A;:H/AKM>/N "O#QPR S.D/T S.DS.4/T+/1:UK/T KM>/12I .4"$/1 #
"O# which in turn implies
" and ,
" , then it implies
,
,
If ,
V &X"W , &K , 27&+ has Y[ZU5 roots.
G]\>^ V _`+aUbIKbcAE , d_fe as 5gih j . (Done)
We know
, & , 27&+ "$# which implies , & , 32I&; "O# .
In other words,
&
&l"W +m<( n < J@oB for general C we have, pS, 32IqlrNK , 2sqlrut0v>(w(w>w> , 2sqlrut7q!xyvK+z .
If ,
, 327qlisrNaK root
, 2sqlofrut0k v>(w(w>w> , 2sqlrut7q!xyvKover
+{8 #
<xyv
@
B
S
>
(
w
>
w
>
w
_fe~} q!xyv of degree }fa .
<
T
<
|
Find roots of
over ? < maximum number of roots
-+< " , 27&DCFEHG % & ,
Lemma 1.2. There exists a irreducible polynomial E(X) such that ,
k S€ needs to be non-zero or [ < does’t divide .
_ degree of Y in+
(1)
degree of Y in
_ eVƒ‚ } by choice of D
(2)
+a
In fact for constant rate code, h „ 6T…†
& " <xyv &2 is irreducible over ? < .
Lemma 1.3. %
ˆ‡‰‹Š .
Exact characterization of irreducible polynomial of the form
&A<Œ8 , for , & ? <Ž ‘
Lemma 1.4. ,
S’”“–•K€ < " ’o < “—•K <
Proof. Hint: Show
Proof. Proof of lemma (1.1) using (1.4)
Done if
&‰ˆ<xyv{ˆ2
True if %
27&˜8 # CFEUG % &
, & < <  ˆ, 27™
8 # CFEHG % &
P
P % &AG 6Nšo6 G]\ … › <Kxyv &2y
1
(3)
(4)
(5)
In for number of folds,
C
general,

5Xg œbc / ž Ÿ v +a“O¡j +
2