Uncorrectable Errors of Weight Half the Minimum Distance for Binary Linear Codes
Kenji Yasunaga* Toru Fujiwara+
*Kwansei Gakuin University, Japan+Osaka University, Japan
2008 IEEE International Symposium on Information Theory, July 6th to 11th, 2008 Sheraton Centre Toronto Hotel, Toronto, Ontario, Canada
2
Summary of the Work
Main Result A lower bound on #(uncorrectable errors of weight ) for
binary linear codes. d : the minimum distance of the code
A generalization to weight .
Main Techniques Monotone error structure (Larger half)
Monotone error structure appears in [Peterson, Weldon, 1972] . Larger half was introduced in [Helleseth, Kløve, Levenshtein, 2005] .
2/d
2/d
3
Outline
Correctable/Uncorrectable Errors
Our Results
Monotone Error Structure
Proof Sketch of Our Results
4
Outline
Correctable/Uncorrectable Errors
Our Results
Monotone Error Structure
Proof Sketch of Our Results
5
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 .
6
Correctable/Uncorrectable Errors
Correctable errors E0(C) = Correctable by minimum distance decoding. Ei
0 (C) : Correctable errors of weight i
Uncorrectable errors E1(C) = {0,1}n 〵 E0(C) Ei
1(C) : Uncorrectable errors of weight i
The error probability over BSCp is .
Minimum distance decoding Outputs a nearest (w.r.t. Hamming dist.) codeword to the input. Performs ML decoding for BSC. Syndrome decoding is a minimum distance decoding.
in
CECE ii |)(||)(| 10
|)(|)1( 1
0
CEppP i
n
i
inierror
7
Syndrome Decoding
Coset partitioning
Syndrome decoding Output y + vi if y∈Ci ( y is the input). Coset leaders = Correctable errors.
jiCCC jii
in
kn
for,}1,0{2
1
)(minarg}:{
vvccvw
CC
iCvi
ii
: Coset of C: Coset leader of Ci
8
Outline
Correctable/Uncorrectable Errors
Our Results
Monotone Error Structure
Proof Sketch of Our Results
9
Previous Results for |Ei1(C)|
For the first-order Reed-Muller code RMm
|Ed/21(RMm)| [Wu, 1998]
|Ed/2+11(RMm)| [Yasunaga, Fujiwara, 2007]
For binary linear codes Upper bounds on |Ei
1(C)| for every 0 ≤ i ≤ n [Poltyrev 1994], [Helleseth, Kløve 1997], [Helleseth, Kløve, Levenshtein 2005]
10
Our Results
A lower bound on for codes satisfying some condition.
The condition is not too restrictive. Long Reed-Muller codes and random linear codes satisfy
Given by #(codewords of weight d (and d+1)). Asymptotically coincides with the corresponding upper bound for
Reed-Muller codes and random linear codes.
A generalization to for . The bound is weak.
|)(| 12/ CE d
|)(| 1 CEi 2/di
11
Outline
Correctable/Uncorrectable Errors
Our Results
Monotone Error Structure
Proof Sketch of Our Results
12
Monotone Error Structure
Recall that a coset leader is a minimum weight vector in a coset.
There may be more than one minimum weight vector 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.
13
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 vectors that are covered by ⇒ v are correctable. v is uncorrectable. All vectors that cover ⇒ v are uncorrectable.
Example 1100 is correctable. ⇒ 0000, 1000, 0100 are correctable. 0011 is uncorrectable. ⇒ 1011, 0111, 1111 are uncorrectable.
14
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) = Uncorrectable errs. that are not covered by other uncorrectable
errs. M1(C) uniquely determines E1(C).
Larger half LH(c) of c∈C Introduced for characterizing M1(C) in [Helleseth et al., 2005]. Combinatorial construction is given in [Helleseth et al., 2005].
M1(C) ⊆ LH(C 〵 {0}) ⊆ E1(C), where .
Sc
LHSLH
)()( c
15
Outline
Correctable/Uncorrectable Errors
Previous Results
Our Results
Monotone Error Structure
Proof Sketch of Our Results
16
Proof Sketch of Our Results
Objective : To derive a lower bound on .
The following equalities hold:
[Proof] Since is the smallest weight in E1(C), uncorrectable errors of
weight do not cover any other uncorrectable errors.
⇒
Derive a lower bound on .
)(}){\()( 12/2/
12/ CECLHCM ddd 0
|)(| 12/ CE d
2/d
)(}){\()( 11 CECLHCM 0
)()( 12/
12/ CECM dd
|}){\(| 2/ 0CLH d
2/d
Proof Sketch of Our Results ( d is even )
, where Ai(C) = { codewords of weight i in C}.
Larger halves of two codewords in Ad(C) are almost disjoint.
))((}){\(2/ CALHCLH dd 0
1|)()(| 21 cc LHLH for every )(, 21 CAdcc
17
|}){\(||)(|)1|)(||)((| 2/ 0CLHCACALH dddi c
|)(| 12/ CEd
2/2
1d
d
= =
LH( ・ )
・・・
}){\(2/ 0CLH d
・・・・・・
・・・
c1
c2
c4
c3
Ad(C)
≤ |Ad(C)| – 1
|LH(ci)|
2/21
dd
For every ci ∈ Ad(C),#(common LH) is less than |Ad(C)|.
The Results ( d is even )
When d is even, if holds, then
If as then upper and lower bounds
asymptotically coincide. For Reed-Muller codes and random linear codes, the upper and lower
bounds asymptotically coincide.
18
1|)(|2/2
1
CA
dd
d
.|)(|2/2
1|)(||)(|)1|)((||)(|2/2
1 12/ CA
dd
CECACACAd
dddddd
02/
|)(|
d
dCAd n
Upper bound is from [Helleseth et al. 2005]
The Results ( d is odd )
19
When d is odd, if holds, then
If as then upper and lower
bounds asymptotically coincide.
n
1|)(||)(|2/)1( 1
CACA
dd
dd
.|)(||)(|2/)1(
|)(|
|)(|)1|)(||)(|2(|)(||)(|2/)1(
11
2/)1(
111
CACAd
dCE
CACACACACAd
d
ddd
ddddd
02/)1(
|)(| 1
dd
CAd
Upper bound is from [Helleseth et al. 2005]
A similar argument can be applied to weight .
For large i The condition for the bound is more restrictive. The bound is weak.
The bound is a lower bound on LHi(C). The difference between LHi(C) and Ei
1(C) is large.
A Generalization to Larger Weights
20
iB
idi
ii
2/23
32
|))(||)((|12
2|)(|32
|)(||)(|2/2
332
21222
1
CACAi
iCA
ii
CECLHBBidi
ii
iii
iiii
For an integer i with ⌈d/2 ≤ ⌉ i ≤ ⌊n/2⌋, if holds, then
2/di
where Bi = |A2i−2(C)| +|A2i−1(C)| + |A2i(C)|.
Conclusion
Main results A lower bound on #(correctable errors of weight ) for
binary linear codes satisfying some condition.
The bound asymptotically coincides with the upper bound for Reed-Muller codes and random linear codes.
Monotone error structure & larger half are main tools. A generalization to weight is also obtained.
The generalized bound is weak for large i .
Future work A good lower bound for weight .
21
2/d
2/di
2/d
Codes Satisfying the Condition
The condition
Codes satisfying the condition (n, k) primitive BCH codes for n = 127 and k ≤ 64, n = 63 and k ≤ 24 (n, k) extended primitive BCH codes for n = 127 and k ≤ 64, n = 63 and
k ≤ 24 r-th order Reed-Muller codes of length 2m
fixed r and m →∞ Random linear codes for n → ∞
22
1|)(|2/2
1
TA
dd
d
1|)(||)(|2/)1( 1
TATA
dd
dd
r m
1 ≥ 4
2 ≥ 6
3 ≥ 8
4 ≥ 10
5 ≥ 11
6 ≥ 13
for even d
for odd d
23
Proof Sketch of Our Results ( d is even )
x2
・・
・
x1
)(2/ CLH d
24
Proof Sketch of Our Results ( d is even )
c1
・・
・
c2
c4
c3
Ad(C)
Ai(C) : the set of codewords with weight i
x1x2
)(2/ CLH d
25
Proof Sketch of Our Results ( d is even )
・・
・
LH( ・ )
・・
・
x1x2c1
c2
c4
c3
Ad(C)
Ai(C) : the set of codewords with weight i
)(2/ CLH d
26
Proof Sketch of Our Results ( d is even )
・・
・
・・
・・
・・
x1x2c1
c2
c4
c3
LH( ・ )
Ad(C)
Ai(C) : the set of codewords with weight i
)(2/ CLH d
27
Proof Sketch of Our Results ( d is even )
・・
・
・・
・・
・・
・・
・
x1x2c1
c2
c4
c3
LH( ・ )
Ad(C)
Ai(C) : the set of codewords with weight i
)(2/ CLH d
28
Proof Sketch of Our Results ( d is even )
・・
・
・・
・・
・・
・・
・・
・・
x1x2c1
c2
c4
c3
LH( ・ )
Ad(C)
Ai(C) : the set of codewords with weight i
)(2/ CLH d
29
Proof Sketch of Our Results ( d is even )
・・
・
・・
・・
・・
・・
・・
・・
|LH(ci)|x1x2c1
c2
c4
c3
LH( ・ )
2/21
dd
Ad(C)
Ai(C) : the set of codewords with weight i
)(2/ CLH d
30
Proof Sketch of Our Results ( d is even )
・・
・
・・
・・
・・
・・
・・
・・
|LH(ci)|
≤ |Ad(T)| – 1
x1x2c1
c2
c4
c3
LH( ・ )
2/21
dd
Ad(C)
Ai(C) : the set of codewords with weight i
)(2/ CLH d
31
Proof Sketch of Our Results ( d is even )
・・
・
・・
・・
・・
・・
・・
・・
For every ci ∈ Ad(T),#(common LH) is less than |Ad(T)| – 1
Thus,if |LH(ci)| > |Ad(T)| – 1, then (|LH(ci)| - |Ad(T)|+1) |Ad(T)| ≤ |LHd/2(T)|
|LH(ci)|
≤ |Ad(T)| – 1
x1x2c1
c2
c4
c3
LH( ・ )
2/21
dd
Ad(C)
Ai(C) : the set of codewords with weight i
)(2/ CLH d
A Generalization to Larger Weight
The lower bound for weight is obtained by considering the vectors of weight in
A similar argument can be applied to weight However, for large i, The condition for the bound is more restrictive The bound is weak
The bound is a lower bound on LHi(C) The difference between LHi(C) and Ei
1(C) is large
32
)()()( 11 CECLHCM
2/d
2/d
12/ di