Correctable Errors of Weight
Half the Minimum Distance Plus One
for the First-Order Reed-Muller Codes
Kenji Yasunaga
Toru Fujiwara
Osaka University, Japan
Applied Algebra, Algebraic Algorithms, and Error Correcting Codes (AAECC-17),
Indian Institute of Science, Bangalore, December 16-20, 2007
Summary of the Work
Main Result
 An explicit expression for #(correctable errors of weight
d/2+1) for the first-order Reed-Muller codes is derived

d : the minimum distance of the code
Main Techniques
 Monotone error structure (Larger half)


Monotone error structure appeared in [Peterson, Weldon, 1972]
Larger half was introduced by [Helleseth, Kløve, Levenshtein,
2005]
2
Outline

Correctable Errors

First-order Reed-Muller Codes

Previous Results

Our Results

Monotone Error Structure

Proof Sketch of Our Results
3
Outline

Correctable Errors

First-order Reed-Muller Codes

Previous Results

Our Results

Monotone Error Structure

Proof Sketch of Our Results
4
Problem Setting

Binary linear code C ⊆ {0,1}n

Error vector e ∊ {0,1}n

If w(e) < d/2 ⇒ e is always correctable
If w(e) ≥ d/2 ⇒ ?
 w(x) : the Hamming weight of x
In this work, we investigate
#(correctable errors of weight i ) for i ≥ d/2
5
Correctable/Uncorrectable Errors

Correctable errors E0(C)
= Correctable by Minimum Distance (MD) decoding
 E0i(C) : Correctable errors of weight i

Uncorrectable errors E1(C) = {0,1}n 〵E0(C)
 E1i(C) : Uncorrectable errors of weight i
n
 | E (C ) |  | E (C ) |   
i
 
0
i

1
i
MD decoding



Output a nearest (w.r.t. Hamming dist.) codeword to the input
Perform ML decoding for binary symmetric channels
Syndrome decoding is an MD decoding
6
Syndrome Decoding

Coset partitioning
2
{0 , 1} 
n
nk

Ci ,
C i  C j   for i  j
i 1
C i  { v i  c : c  C } : Coset of
v i  arg min w ( v )
vC i

C
: Coset leader of Ci
Syndrome decoding



Output y + vi if y∈Ci ( y is the input)
Coset leaders = Correctable errors
Perform MD decoding
7
Outline

Correctable Errors

First-order Reed-Muller Codes

Previous Results

Our Results

Monotone Error Structure

Proof Sketch of Our Results
8
First-Order Reed-Muller Code


RMm : The first-order Reed-Muller code of length 2m

Dimension = m+1

Minimum distance d = 2m-1
RMm ⇔ Linear Boolean functions with m variables
|E0i(RMm)|×2m+1 = #(Boolean func. of nonlinearity i )

Nonlinearity of Boolean function f
 Distance between f and linear Boolean functions
 Important
criteria for cryptographic applications [Canteaut,
Carlet, Charpin, Fontaine, 2001]
9
Outline

Correctable Errors

First-order Reed-Muller Codes

Previous Results

Our Results

Monotone Error Structure

Proof Sketch of Our Results
10
Previous Results for |E0(RMm)|

[Berlekamp, Welch, 1972]



The weight distributions of all cosets of RM5
⇒ |E0i(RM5)| for all 0 ≤ i ≤ n
By computer
[Wu, 1998]

An explicit expression for |E0d/2(RMm)|

By revealing the structure of coset leaders of weight d/2
1. Coset leaders of weigh d/2 ⇒ 3 types
2. Determine #(coset leaders) for each type
11
Outline

Correctable Errors

First-order Reed-Muller Codes

Previous Results

Our Results

Monotone Error Structure

Proof Sketch of Our Results
12
Our Results

An explicit expression for |E0d/2+1(RMm)|

By using the monotone error structure (Larger half)
 Monotone
error structure appeared in [Peterson, Weldon,
1972]
 Larger half was introduced by [Helleseth, Kløve, Levenshtein
2005]

Lead to #(Boolean functions of nonlinarity d/2+1)

Compared to [Wu, 1998],
 Our
approach does not fully reveal the structure of coset
leaders of weight d/2+1
 Our
approach can give a simpler proof for |E0d/2(RMm)|
13
Outline

Correctable Errors

First-order Reed-Muller Codes

Previous Results

Our Results

Monotone Error Structure

Proof Sketch of Our Results
14
Monotone Error Structure

Recall that a coset leader is a minimum weight vector in
a coset

There may be one more minimum weight vectors in the
same coset
⇒ Any of them will do

If we take the lexicographically smallest one for all
cosets,
⇒ Correctable/uncorrectable errors have a monotone
structure
15
Monotone Error Structure

Notation


Support of v : S(v) = { i : vi≠0 }
v is covered by u : S(v) ⊆ S(u)

Monotone error structure
v is correctable
⇒ all u s.t. S(v) ⊆ S(u) are correctable
v is uncorrectable
⇒ all u s.t. S(u) ⊇ S(v) are uncorrectable

Example


1100 is correctable
⇒ 0000, 1000, 0100 are correctable
0011 is uncorrectable ⇒ 1011, 0111, 1111 are uncorrectable
16
Minimal uncorrectable errors

Errors have the monotone structure (w.r.t ⊆)
⇒ E1(C) is characterized by minimal vectors (w.r.t. ⊆)

Minimal uncorrectable errors M1(C)
1
 = Minimal vectors (w.r.t. ⊆) in E (C)
1
1
 M (C) uniquely determines E (C)

Larger half LH(c) of c∈C

Introduced for characterizing M1(C) in [HKL2005]

Combinatorial construction is given in [HKL2005]

M1(C) ⊆ LH(C〵{0}), where LH ( S )   LH ( c )
c S
17
Outline

Correctable Errors

First-order Reed-Muller Codes

Previous Results

Our Results

Monotone Error Structure

Proof Sketch of Our Results
18
Proof Sketch of Our Results

We will determine |E1d/2+1(RMm)|

Observe the relations between E1d/2(RMm), E1d/2+1(RMm),
LH(RMm〵{0, 1}), M1(RMm)
E1d/2(RMm)
E1d/2+1(RMm)
19
Proof Sketch of Our Results

We will determine |E1d/2+1(RMm)|

Observe the relations between E1d/2(RMm), E1d/2+1(RMm),
LH(RMm〵{0, 1}), M1(RMm)
E1d/2(RMm)
E1d/2+1(RMm)
LH(RMm〵{0, 1})
LH(RMm〵{0, 1}) ⊆ E1d/2(RMm) ∪ E1d/2+1(RMm)
20
Proof Sketch of Our Results

We will determine |E1d/2+1(RMm)|

Observe the relations between E1d/2(RMm), E1d/2+1(RMm),
LH(RMm〵{0, 1}), M1(RMm)
E1d/2(RMm)
E1d/2+1(RMm)
LH(RMm〵{0, 1})
M1(RMm)
M1(RMm) ⊆ LH(RMm〵{0, 1}) ⊆ E1d/2(RMm) ∪ E1d/2+1(RMm)
21
Proof Sketch of Our Results

Consider Wm = { v : S(v)⊆S(c) for c∈RMm〵{0,1}, w(v)=d/2+1}
E1d/2(RMm)
E1d/2+1(RMm)
Wm
LH(RMm〵{0, 1})
M1(RMm)
22
Proof Sketch of Our Results

Consider Wm = { v : S(v)⊆S(c) for c∈RMm〵{0,1}, w(v)=d/2+1}
E1d/2(RMm)
Wm
E1d/2+1(RMm)
|Wm| is
easily obtained
LH(RMm〵{0, 1})
M1(RMm)
23
Proof Sketch of Our Results

Consider Wm = { v : S(v)⊆S(c) for c∈RMm〵{0,1}, w(v)=d/2+1}
E1d/2(RMm)
Wm
LH(RMm〵{0, 1})
M1(RMm)
E1d/2+1(RMm)
|Wm| is
easily obtained
We determine
this size
24
Proof Sketch of Our Results
E1d/2(RMm)
Wm
LH(RMm〵{0, 1})
E1d/2+1(RMm)
|Wm| is
easily obtained
We determine
this size
M1(RMm)

Observe that the vectors v in
are non-minimal
⇒ v is obtained by adding a weight-one vector
to a minimal uncorrectable error
25
Proof Sketch of Our Results
E1d/2(RMm)
Wm
LH(RMm〵{0, 1})
M1(RMm)

E1d/2+1(RMm)
|Wm| is
easily obtained
We determine
this size
Vm
Observe that the vectors v in
are non-minimal
⇒ v is obtained by adding a weight-one vector
to a minimal uncorrectable error
⇒ Construct such a set Vm
and determine |Vm〵Wm |
26
The Results

For m ≥ 5,
|E
1
d / 2 1
( RM
) | 4 ( 2  1)( 2
m
m
m 3
m
 2 m 1 

2
  ( 4 m  2  3)
 1)  m  2
2

 3

1







m


2
0
1


 | E d / 2  1 ( RM m ) |  | E d / 2  1 ( RM m ) |   m  2

2

1


27
Conclusions

#(correctable errors of weight d/2+1) is derived
for the first-order Reed-Muller codes


Monotone error stucture & larger half are main tools
Our approch does not reveal the structure of coset
leaders of weight d/2+1
 [Wu,
1998] reveals the structure of coset leaders of weight
d/2 to derive #(correctable errors of weight d/2)
28