Limits to List Decoding
Reed-Solomon Codes
Venkatesan Guruswami
Atri Rudra
(University of Washington)
May 24, 2005
STOC 2005, Baltimore
1
Error-Correcting Codes


Linear Code C : GF(q)k! GF(q)n
Hamming Distance or (u,v) for u,v 2 GF(q)n



Distance of code C, d=minx,y2GF(qk) (C(x),C(y))
C is an [n,k,d]GF(q) code


Number of positions u and v differ
Relative distance =d/n
This talk is about Reed-Solomon (RS) Codes
May 24, 2005
STOC 2005, Baltimore
2
List Decoding

Given r 2 GF(q)n and 0· e· 1


Output all codewords c 2 C such that (c,r)· en
Combinatorial Issues

How big can the list of codewords be ?
LDR(C) largest e such that list size is poly(n)
Algorithmic issues




Can one find list of codewords in poly(n) time ?
Cannot have poly time algo beyond LDR(C) errors
May 24, 2005
STOC 2005, Baltimore
3
List Recovery





Related to List Decoding Problem
Given C: GF(q)k! GF(q)n and Liµ GF(q), 1· i· n
Find all codewords c=hc1,,cni s.t. ci2 Li 8 i
|Li|· s
LRB(C) largest s for which # of codewords
is poly(n)
1
L1
May 24, 2005
2
L2
3
n
L3
Ln
STOC 2005, Baltimore
4
Reed-Solomon Codes

RS [n,k+1]GF(q)







Message P a poly. of degree · k over GF(q)
S µ GF(q)
RS(P) = h P(a) ia2S
n= |S|
d=n–k
For this talk S = GF(q)
9 poly time algo for list decoding of RS codes
till error bound J()=1-(1-)1/2=1- (k/N)1/2
May 24, 2005
STOC 2005, Baltimore
5
The Big Picture for RS
9 poly time algo for error bound · J()
Polynomial Reconstruction
RS List Recovery
RS List Decoding
Negative Result ) above algo optimal
May 24, 2005
STOC 2005, Baltimore
6
Talk outline

Our main result is about combinatorial
limitation of List Recovery of Reed Solomon
Codes

Motivation of the problem
Main Result and Implications
Proof of the main result


May 24, 2005
STOC 2005, Baltimore
7
Combinatorial Limitations- I

Half Distance
For any C

Unique decodability
Error Bound

LDR(C ) ¸ /2
Relative Distance ()
May 24, 2005
STOC 2005, Baltimore
8
Combinatorial Limitations- II
Half Distance

Full Distance
LDR(C)¼  is the best one
can hope for
Error Bound


Lots of “good” codes with
LDR(C)¼ 


Relative Distance ()
May 24, 2005
e ¸  can’t detect errors

STOC 2005, Baltimore
Random Linear Codes
2x improvement over unique
decoding
Difficulty: getting explicit codes
9
Combinatorial Limitations- III
Half Distance
Full Distance

Johnson Bound
Johnson Bound

Error Bound


For any code C
LDR(C) ¸ J()=(1-(1-)1/2)
Exists codes for which
Johnson Bound is tight


Non-linear codes [GRS00]
Linear codes [G02]
Relative Distance ()
May 24, 2005
STOC 2005, Baltimore
10
Going beyond the Johnson Bound
Half Distance
Full Distance
Johnson Bound

Go beyond Johnson
Bound
Error Bound




Choice of code matters
Random Linear codes get
there
What about well studied
codes like RS codes ?
Motivation of our work
Relative Distance ()
May 24, 2005
STOC 2005, Baltimore
11
Algorithmic Status of RS
Half Distance
Full Distance

Johnson Bound
Unique decoding

?

?

?
?
?
May 24, 2005
??
List Decoding

?

[Peterson60]
Johnson Bound
[Sud97, GS99]
Unknown beyond JB

STOC 2005, Baltimore
Some belief that
LDR(RS)=(1-(1-)1/2)
12
General setup for GS algorithm

Polynomial Reconstruction




Pairs of numbers {(ai,bi)}, i=1..N
Finds all degree k poly P at most N-(Nk)1/2
indices i, P(ai)  bi
ai distinct ) List Decoding of RS
ai not necessarily distinct ) List Recovery of RS
a1 a2 a3
b b2 b3
1
ai
bi
an
bn
Li
May 24, 2005
STOC 2005, Baltimore
13
Main Result of this talk

Version of Johnson Bound implies
LRB(RS) ¸ dn/ke -1
(GS algo works in poly time in this regime)
We show LRB(RS) = dn/ke-1
) For Polynomial reconstruction GS algo is
optimal
May 24, 2005
STOC 2005, Baltimore
14
Implication for List Decoding RS
a1
a2
a3
ai
an
dn/ke


Polynomial Reconstruction

In List Recovering setting N=n¢dn/ke

Number of disagreements = N-n w (Nk)1/2
With (little more than) N-(Nk)1/2 disagreement have
super poly RS codewords


GS algo works for disagreement · N-(Nk)1/2
Improvement “must” use near distinctness of ais
May 24, 2005
STOC 2005, Baltimore
15
Main Result
a1
a2
a3
ai
an
dn/ke



n=q=pm
m
m-1+pm-2++p+1
 D=(p -1)/(p-1) =p
Consider RS [n,k=D+1]GF(qm)*
For each i=1,,n the list Li= GF(p)
 dn/ke = p
Number of deg D polys over GF(pm) which take
m
2
values in GF(p) is p
May 24, 2005
STOC 2005, Baltimore
16
Explicit Construction of Polys



Pb(z) = 
zai + 1)D where bi2 GF(p)
 a is a generator of GF(pm)
 D=(pm-1)/(p-1)=pm-1++p+1
Poly over GF(pm)
2m-1
i=0 bi(
Takes values in GF(p)
 Norm function: for all x2 GF(pm), xD2 GF(p)
Will now prove for distinct b, Pb(z) are
distinct polys over GF(pm)
May 24, 2005
STOC 2005, Baltimore
17
Proof Idea

By Linearity,need to show
Pb(z) = 
bi( zai + 1)D  0
) b1 = b2 == b2m-1 =0
2m-1
i=0

Coefficients of all zj must be 0
( )

D=pm-1++p+1
D
2m-1
j
i=0
bi (ai)j =0 for j=0..D
D+1 eqns and 2m vars (some of them trivial)
May 24, 2005
STOC 2005, Baltimore
18
Lucas’ Lemma

p prime and integers a and b


a=a0+a1p++arpr
b=b0+b1p++brpr
( )  ( ) ( )( ) mod p
a
b

a0
b0
a1
b1
ar
br
D=1+p+ pm-1, j=j0+j1p+ +jmpm-1
( )  0 iff for all i, j 2 {0,1}
D
j
m
i
m
) 2 equations and 2 var
May 24, 2005
STOC 2005, Baltimore
19
Wrapping up the proof

2m equations in 2m variables
 T= { j0+j1p+ jm-1pm-1 | ji2 {0,1} }

2m-1
i=0
bi (ai)j =0 for j 2 T
Coefficient matrix is Vandermonde
1 aj0 (aj0)2 … (aj0)2
1
.
.
.
1
May 24, 2005
m-1
aj1 (aj1)2 … (aj1)2m-1
b0
b1
STOC 2005, Baltimore
=0
20
Other Results in the Paper

Use connection with BCH codes to get an
exact estimate

Show existence of explicit received word with
super poly “close by” RS codewords for
certain parameters

Uses ideas from [CW04]
May 24, 2005
STOC 2005, Baltimore
21
Open Questions

Is Johnson Bound the true list decoding
radius of Reed Solomon codes ?

Show RS of rate 1/L cannot be list recovered
using lists of size L which are not prime
powers.

What RS codes on prime fields ?
May 24, 2005
STOC 2005, Baltimore
22