Optimal Conflict-avoiding Codes of Odd Length Weight Three

Optimal Conflict-avoiding Codes of Odd Length Weight

Three

Department of Applied MathematicsNational Chiao Tung University, Taiwan

Yuan-Hsun Lo ( 羅元勳 )

A joint work with Kenneth Shum and Hung-Ling Fu

2

Definition

Conflict-avoiding code CAC(n,k)

Length n

Hamming weight k

Inner product of arbitrary cyclic shift of any two distinct sequences is either 0 or 1.

(1 0 0 1 0)

(1 1 0 0 0)

3

Multiple-access collision channel without feedback

Application

M potential users.

When more than one users transmit packets at the same time, a conflict (collision) occurs.

Arbitrary active time slot. At most k users are active at the same time.

Inactive → active : at least n time slots.Guarantee: every active user can transmit at least one packet successfully in a frame of n

slots.

Guarantee: every active user can transmit at least one packet successfully in a frame of n

slots.

4

Image of Usage

Senders

B

a = (1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0)

b = (0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0)

c = (0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0)

d = (0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1)

A

C

D

Receivers

B’

A’

C’

D’

Time Slots

M = 4, n = 17, k = 3

CAC(17,3)

5

Image of Usage

Senders

B

a = (1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0)

b = (0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0)

c = (0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0)

d = (0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1)

A

C

D

Receivers

B’

A’

C’

D’

Time Slots

M = 4, n = 17, k = 3

CAC(17,3)

6

Image of Usage

Senders

B

a = (1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0)

b = (0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0)

c = (0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0)

d = (0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1)

A

C

D

Receivers

B’

A’

C’

D’

Time Slots

M = 4, n = 17, k = 3

CAC(17,3)

7

Image of Usage

1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Senders

B

a = (1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0)

b = (0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0)

c = (0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0)

d = (0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1)

A

C

D

Receivers

B’

A’

C’

D’

Time Slots

M = 4, n = 17, k = 3

CAC(17,3)

8

Image of Usage

Senders

B

a = (1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0)

b = (0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0)

c = (0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0)

d = (0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1)

A

C

D

Receivers

B’

A’

C’

D’

Time Slots

M = 4, n = 17, k = 3

CAC(17,3)

9

Image of Usage

Senders

B

a = (1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0)

b = (0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0)

c = (0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0)

d = (0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0)

A

C

D

Receivers

B’

A’

C’

D’

Time Slots

M = 4, n = 17, k = 3

CAC(17,3)

10

Image of Usage

Senders

B

a = (1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0)

b = (0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0)

c = (0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0)

d = (0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0)

A

C

D

Receivers

B’

A’

C’

D’

Silence SymbolSurvived PacketCollided Packet

Time Slots

M = 4, n = 17, k = 3

CAC(17,3)

11

Objective

Given n and k, maximize M.

Optimal CAC : a CAC with maximum size

M(n, k): the size of an optimal CAC(n, k)

12

Outline

1. Review of the literature of CAC

2. Formulation using Graph Theory

3. Some new optimal CAC of weight 3

and odd length.

13

Outline




and odd length.

14

Optimal CAC of weight 3

Theorem (Levenshtein and Tonchev, 2005)

For n ≡ 2 (mod 4), then M(n, 3) = (n – 2)/4.

For n is odd, then M(n, 3) ~ n/4 as n → ∞.

15

Optimal CAC of weight 3

Theorem (Jimbo et al., 2007)

Let n = 4t. Then 7 / 32 if 0 (mod 8),

(7 4) / 32 if 1 (mod 8),

(7 24) / 32 if 2,10 (mod 24),

(7 12) / 32 if 3 (mod 24),

(7 16) / 32 if 4,20 (mod 24),

(7 12) / 32 if 5,13 (mo( ,3)

n t

n t

n t

n t

n t

n tM n

d 24),

(7 8) / 32 if 6 (mod 8),

(7 4) / 32 if 7 (mod 8),

(7 20) / 32 if 11,19 (mod 24),

(7 16) / 32 if 12 (mod 24),

(7 8) / 32 if 18 (mod 24),

(7 20) / 32 if 21 (mod 24).

n t

n t

n t

n t

n t

n t

Jimbo et al., 2007 →

Mishima et al., 2009 →

Fu, Lin and Mishima, 2010 →

16

CAC of weight > 3

Some constructions of optimal CAC of weight 4 and 5 Momihara, Jimbo et al. (2007)

For general weight Kenneth and Wong (2010)

17

CAC of weight > 3

Some constructions of optimal CAC of weight 4 and 5 Momihara, Jimbo et al. (2007)

For general weight Kenneth and Wong (2010)

( , ) 1limsup

2 2n

M n k

n k

18

We are interested in odd n and k = 3.

We are interested in odd n and k = 3.

19

Outline



– set representation

– hypergraph matching


and odd length.

20

Set Representation

We can use subsets of to represent codewords by their natural correspondence.

The difference set of a codeword is defined by Δ(x) = {i – j (mod n) : i, j ∈ x, i≠j}.

nx

n

x = (1 1 0 1 0 0 0 0 0 0 0 0 0 )±1 ±2

±3

Δ(x) = {±1, ±2, ±3} = {1, 2, 3, 10, 11, 12}

x = {0, 1, 3}

0 1 2 3 4 5 6 7 8 9 10 11 12

Example (n = 13, k = 3)

21

Set Representation

The difference set from a codeword x can be redefined as:

Δ(x) = {i – j ≤ n/2 : i, j ∈ x, i≠j}

By cyclically shifting the codeword, we can assume without loss generality that 0 ∈ x for any codeword x.

22

Equivalent Definition of CAC

A CAC (n, 3) is a collection of 3-subsets of such that Δ(x) ∩ Δ(y) = ψ for x ≠ y

n

23

Equivalent Definition of CAC

A CAC (n, k) is a collection of k-subsets of such that Δ(x) ∩ Δ(y) = ψ for x ≠ y

( ) { 1, 2, 3, , / 2 }x C

x n

n

Packing {1, 2, …, n/2} to obtain as many codewords

as possible (optimal CAC).

|Δ(x)| is as small as possible

24

Equi-difference Codewords

A codeword of form {0, i ,2i} is said to be

equi-difference.

Example (n = 15, k =3)

x = {0, 5, 10}

y = {0, 4, 8 }

z = {0, 7, 9 }

→ Δ(x) = {5}

→ Δ(y) = {4, 7}

→ Δ(z) = {2, 6, 7}

equi-difference codewords

25

Let x be a codeword of a CAC (n, 3).

1) If Δ(x) = {i}, then i = n/3.

Characterization of Δ

0 n/3 2n/3

26


1) If Δ(x) = {i}, then i = n/3.

2) If Δ(x) = {i, j}, then j ≡ ±2i (mod n).


0 i 2ij

ii

0 i 2i

ii

j

27


1) If Δ(x) = {i}, then i = n/3.

2) If Δ(x) = {i, j}, then j ≡ ±2i (mod n).

3) If Δ(x) = {i, j, k}, then i + j ≡ ±k (mod n).


0 i i+jk

i j

0 i i+j k

i j

28

Graphical Characterization

H(n): a hypergraph (V, E)

V: vertex set {1, 2, 3, …, (n –1)/2 }

(the set of differences arising from codewords)

E: hyperedge set such that e E if e can correspond to a codeword. (|e| = 1, 2 or

3 )

An optimal CAC corresponds to

a maximum hypergraph matching.

29


G(n): a graph obtained from H(n) by

dropping all hyperedges with size 3

Each edge of G(n) corresponds to

an equi-difference codeword.

In G(n), i ~ j iff i ≣ ±2j (mod n).

30

G(n) is 2-regular (i.e., a union of cycles). G(n) contains at most 1 loop.


1 2 4 3 5G(11) :

1 2 4 8 3G(17) :

6 5 7

G(21) :

1 2 4 8 5 10

3 6 9 7

i ~ j iff i ≣ ±2j (mod n)

31


G(21) :

1 2 4 8 5 10

3 6 9 7

Δ = {7} → {0, 7, 14} → 100000010000001000000

Δ = {1, 2} → {0, 1, 2} → 111000000000000000000

Δ = {4, 8} → {0, 4, 8} → 100010001000000000000

Δ ={5, 10} →{0, 5, 10}→ 100001000010000000000

Δ = {6, 9} →{0, 6, 12}→100000100000100000000

M(21,3) = 5

3

32

Strategy

G(n):

even cycles

odd cycles

33

Another Example: CAC(31,3)

M(31,3) = 7

8 15

41

12

214 3 5 10

13 11

7 6

9

{0,2,5}

Look for a hyperedgewhich intersects three distinct odd cycles

{0,15,30}{0,4,8}

{0,10,20}

{0,9,18}{0,6,12}

{0,7,14}

34

Natural Bounds O(n) = number of odd cycles in G(n)

1 if 3 |

0 if 3 |n

n

n

35

Natural Bounds O(n) = number of odd cycles in G(n)

Theorem 1

For any odd integer n,

1 1( ( ))

2 21 1

( ( ))2

( ,3)

( )

32 n

n

n

nO n

n O nO

M

n

n

1 if 3 |

0 if 3 |n

n

n

36

More Examples CAC(81, 3)

G(34) :

3 6 12 24 33 15

9 18 3627

30 21 39

1 4

2 8

16 17

32 34

13 29

26 23

35

11

40 10

20 5

38 31

19 25

28 7

14 37

22

There is no hyperedges lying across distinct odd cycles.

9a

3b

c

37


G(34) :

3 6 12 24 33 15

9 18 3627

30 21 39

1 4

2 8

16 17

32 34

13 29

26 23

35

11

40 10

20 5

38 31

19 25

28 7

14 37

22


9a

3b

c

M(81,3) = 19

38


G(34) :

3 6 12 24 33 15

9 18 3627

30 21 39

1 4

2 8

16 17

32 34

13 29

26 23

35

11

40 10

20 5

38 31

19 25

28 7

14 37

22


9a

3b

c

M(81,3) = 19

1(3 ,3) 3 2 3

4r rM r

39

Optimal CACs for prime power

Theorem 2

Let p > 3 be a non-Wieferich prime. Then

for r ≥ 1,

1( ,3) 1 ( ,3)

4.r rM p p pr r rM p

40

Optimal CACs for prime power

1(7,3) 1 (7 ,3) 7 2 1

4r rM M r

Theorem 2

Let p > 3 be a non-Wieferich prime. Then

for r ≥ 1,

1( ,3) 1 ( ,3)

4.r rM p p pr r rM p

1(17,3) 4 (17 ,3) 17 1

4r rM M

1(31,3) 7 (31 ,3) 31 2 1

4r rM M r

41

Wieferich prime

Define en = min{e : 2e ≣ 1 (mod n)}.

p is a Wieferich prime if

Only two Wieferich primes, 1093 and 3511, are discovered so far.

The third smallest one > 6.7×1015 if it exists.

2 ppe e

42

Conclusion

If we can find (O(n)–ξn) / 3 mutually disjoint hyperedges of size 3 lying across distinct odd cycles, then equality holds.

1 1( ,3) ( ( ))

)

32 2

(n

nO nnM n O n

43

Conclusion

If we can find (O(n)–ξn) / 3 mutually disjoint hyperedges of size 3 lying across distinct odd cycles, then equality holds.

M(p, 3) is unknown for general p > 3.

1 1( ,3) ( ( ))

)

32 2

(n

nO nnM n O n

1( ,3) 1 ( ,3)

4r rM p p pr r rM p

Conjecture.

There are O(p)/ 3 mutually disjoint phyeredges

lying across distinct odd cycles if O(p) ≥ 3.

44

References• V. I. Levenshtein and V. D. Tonchev, Optimal conflict-avoiding

codes for three active users, In Proc. IEEE Int. Symp. Theory, 2005.

• M. Jimbo et al., On conflict-avoiding codes of length n = 4m for tthree active users, IEEE Trans. Inf. Theory, 2007.

• M. Mishima et al., Optimal conflict-avoiding codes of length n ≣ 0 (mod 16) and weight 3, Des. Codes Cryptogr., 2009.

• H. L. Fu et al., Optimal conflict-avoiding codes of even length and weight 3, IEEE Trans. Inf. Theory, 2010.

• K. Momihara et al., Constant weight conflict-avoiding codes, SIAM J. Discrete Math., 2007.

• K. W. Shum and W. S. Wong, A tight asymptotic bound on the size of constant-weight conflict-avoiding codes, Des. Codes Cryptogr., 2010.

• F. G. Dorais and D. W. Klyve, A Wieferich prime search up to 6.7×1015, J. Integer Seq. 2011.

45

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Optimal Conflict-avoiding Codes of Odd Length Weight Three