Optimal conflict avoiding codes of odd length weight three
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Yuan-Hsun Lo ( 羅 元 勳 ). Optimal Conflict-avoiding Codes of Odd Length Weight Three. Department of Applied Mathematics National Chiao Tung University, Taiwan. A joint work with Kenneth Shum and Hung-Ling Fu. ( 1 0 0 1 0). ( 1 1 0 0 0). Definition. Conflict-avoiding code CAC( n , k )

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Optimal Conflict-avoiding Codes of Odd Length Weight Three

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Optimal conflict avoiding codes of odd length weight three

Yuan-Hsun Lo (羅 元 勳)

Optimal Conflict-avoiding Codes of Odd Length Weight Three

Department of Applied Mathematics

National Chiao Tung University, Taiwan

A joint work with Kenneth Shum and Hung-Ling Fu


Definition

(1 0 0 1 0)

(11 0 0 0)

Definition

Conflict-avoiding codeCAC(n,k)

  • Length n

  • Hamming weight k

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


Application

Application

Multiple-access collision channelwithout feedback

  • M potential users.

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

  • Arbitraryactive 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.


Image of usage

Image of Usage

M = 4,n = 17, k = 3

Senders

Receivers

A

A’

B

B’

C

C’

D

D’

Time Slots

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

CAC(17,3)

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)


Image of usage1

Image of Usage

M = 4,n = 17, k = 3

Senders

Receivers

A

A’

B

B’

C

C’

D

D’

Time Slots

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

CAC(17,3)

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)


Image of usage2

Image of Usage

M = 4,n = 17, k = 3

Senders

Receivers

A

A’

B

B’

C

C’

D

D’

Time Slots

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

CAC(17,3)

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)


Image of usage3

Image of Usage

M = 4,n = 17, k = 3

Senders

Receivers

A

A’

B

B’

C

C’

D

D’

Time Slots

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

CAC(17,3)

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)


Image of usage4

Image of Usage

M = 4,n = 17, k = 3

Senders

Receivers

A

A’

B

B’

C

C’

D

D’

Time Slots

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

CAC(17,3)

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)


Image of usage5

Image of Usage

M = 4,n = 17, k = 3

Senders

Receivers

A

A’

B

B’

C

C’

D

D’

Time Slots

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

CAC(17,3)

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)


Image of usage6

Image of Usage

Silence Symbol

M = 4,n = 17, k = 3

Survived Packet

Collided Packet

Senders

Receivers

A

A’

B

B’

C

C’

D

D’

Time Slots

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

CAC(17,3)

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)


Objective

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)


Outline

Outline

  • Review of the literature of CAC

  • Formulation using Graph Theory

  • Some new optimal CAC of weight 3 and odd length.


Outline1

Outline

  • Review of the literature of CAC

  • Formulation using Graph Theory

  • Some new optimal CAC of weight 3 and odd length.


Optimal cac of weight 3

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→ ∞.


Optimal cac of weight 31

Jimbo et al., 2007 →

Mishima et al., 2009 →

Fu, Lin and Mishima, 2010 →

Optimal CAC of weight 3

Theorem(Jimbo et al., 2007)

Let n = 4t. Then


Cac of weight 3

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)


Cac of weight 31

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)


Optimal conflict avoiding codes of odd length weight three

We are interested in odd n and k = 3.


Outline2

Outline

  • Review of the literature of CAC

  • Formulation using Graph Theory– set representation– hypergraph matching

  • Some new optimal CAC of weight 3 and odd length.


Set representation

±1

±2

±3

Set Representation

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

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

Example (n = 13, k = 3)

x = (11 0 1 0 0 0 0 0 0 0 0 0 )

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

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

x = {0, 1, 3}


Set representation1

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 ∈xfor any codeword x.


Equivalent definition of cac

Equivalent Definition of CAC

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


Equivalent definition of cac1

Equivalent Definition of CAC

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

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

as possible (optimal CAC).

|Δ(x)| is as small as possible


Equi difference codewords

Equi-difference Codewords

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

equi-difference.

Example (n = 15, k =3)

equi-difference codewords

x = {0, 5, 10}

y = {0, 4, 8 }

z = {0, 7, 9 }

→Δ(x) = {5}

→ Δ(y) = {4, 7}

→ Δ(z) = {2, 6, 7}


Characterization of

0

n/3

2n/3

Characterization of Δ

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

  • If Δ(x) = {i}, then i = n/3.


Characterization of1

i

i

0

i

2i

i

i

j

0

i

2i

j

Characterization of Δ

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

  • If Δ(x) = {i}, then i = n/3.

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


Characterization of2

i

j

k

i

j

0

i

i+j

0

i

i+j

k

Characterization of Δ

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

  • If Δ(x) = {i}, then i = n/3.

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

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


Graphical characterization

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 cancorrespond to a codeword. (|e| = 1, 2 or 3 )

An optimal CACcorresponds to

a maximum hypergraph matching.


Graphical characterization1

Graphical Characterization

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

dropping all hyperedges with size 3

In G(n), i ~ j iff i ≣ ±2j (mod n).

Each edge of G(n) corresponds to

an equi-difference codeword.


Graphical characterization2

G(11) :

1

2

4

3

5

G(17) :

1

2

4

8

3

6

5

7

1

2

4

8

5

10

G(21) :

3

6

9

7

Graphical Characterization

  • G(n) is 2-regular (i.e., a union of cycles).

  • G(n) contains at most 1 loop.

i ~ j iff i ≣ ±2j (mod n)


Graphical characterization3

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

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

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

1

2

4

8

5

10

G(21) :

3

6

9

7

Graphical Characterization

3

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

M(21,3) = 5

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


Strategy

Strategy

G(n):

even cycles

odd cycles


Another example cac 31 3

Another Example: CAC(31,3)

{0,4,8}

{0,15,30}

15

8

4

1

{0,7,14}

{0,2,5}

2

14

3

10

{0,10,20}

5

7

6

13

11

12

{0,9,18}

9

{0,6,12}

Look for a hyperedgewhich intersects three distinct odd cycles

M(31,3) = 7


Natural bounds

Natural Bounds

  • O(n) = number of odd cycles in G(n)


Natural bounds1

Natural Bounds

  • O(n) = number of odd cycles in G(n)

Theorem 1

For any odd integer n,


More examples cac 81 3

More Examples CAC(81, 3)

9a

G(34) :

27

9

18

36

3b

30

21

39

3

6

12

24

33

15

16

17

13

29

35

1

4

c

32

34

26

23

11

2

8

38

31

28

7

22

40

10

20

5

19

25

14

37

There is no hyperedges lying across distinct odd cycles.


More examples cac 81 31

More Examples CAC(81, 3)

M(81,3) = 19

9a

G(34) :

27

9

18

36

3b

30

21

39

3

6

12

24

33

15

16

17

13

29

35

1

4

c

32

34

26

23

11

2

8

38

31

28

7

22

40

10

20

5

19

25

14

37

There is no hyperedges lying across distinct odd cycles.


More examples cac 81 32

More Examples CAC(81, 3)

M(81,3) = 19

9a

G(34) :

27

9

18

36

3b

30

21

39

3

6

12

24

33

15

16

17

13

29

35

1

4

c

32

34

26

23

11

2

8

38

31

28

7

22

40

10

20

5

19

25

14

37

There is no hyperedges lying across distinct odd cycles.


Optimal cacs for prime power

Optimal CACs for prime power

Theorem 2

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

for r ≥ 1,


Optimal cacs for prime power1

Optimal CACs for prime power

Theorem 2

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

for r ≥ 1,


Wieferich prime

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.


Conclusion

Conclusion

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


Conclusion1

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.

Conjecture.

There are O(p)/ 3 mutually disjoint phyeredges

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


References

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.


Thank you for your attention

Thank you for your attention


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