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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
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)
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.
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