Uncorrectable Errors of Weight Half the Minimum Distance for Binary Linear Codes

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


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 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


Our Results



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


Our Results



12


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.

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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.

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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


Previous Results

Our Results



16


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.

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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 )

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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 .

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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 → ∞

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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

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x2

・・

・

x1

)(2/ CLH d

24


c1

・・

・

c2

c4

c3

Ad(C)

Ai(C) : the set of codewords with weight i

x1x2

)(2/ CLH d

25


・・

・

LH( ・ )

・・

・

x1x2c1

c2

c4

c3

Ad(C)


)(2/ CLH d

26


・・

・

・・

・・

・・

x1x2c1

c2

c4

c3

LH( ・ )

Ad(C)


)(2/ CLH d

27


・・

・

・・

・・

・・

・・

・

x1x2c1

c2

c4

c3

LH( ・ )

Ad(C)


)(2/ CLH d

28


・・

・

・・

・・

・・

・・

・・

・・

x1x2c1

c2

c4

c3

LH( ・ )

Ad(C)


)(2/ CLH d

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・・

・

・・

・・

・・

・・

・・

・・

|LH(ci)|x1x2c1

c2

c4

c3

LH( ・ )

2/21

dd

Ad(C)


)(2/ CLH d

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・・

・

・・

・・

・・

・・

・・

・・

|LH(ci)|

≤ |Ad(T)| – 1

x1x2c1

c2

c4

c3

LH( ・ )

2/21

dd

Ad(C)


)(2/ CLH d

31


・・

・

・・

・・

・・

・・

・・

・・

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)


)(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

Uncorrectable Errors of Weight Half the Minimum Distance for Binary Linear Codes

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