1. Optimal Conflict-avoiding Codes of Odd Length Weight Three
Yuan-Hsun Lo (羅 元 勳)
Department of Applied Mathematics
National Chiao Tung University, Taiwan
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.
( )
( )

3. Application Multiple-access collision channel without feedback
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.

4. Image of Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ B
Time Slots
a = ( )
CAC(17,3)
b = ( )
c = ( )
d = ( )

5. Image of Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ B
Time Slots
a = ( )
CAC(17,3)
b = ( )
c = ( )
d = ( )

6. Image of Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ B
Time Slots
a = ( )
CAC(17,3)
b = ( )
c = ( )
d = ( )

7. Image of Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ BA A’ B B’ C C’ D D’
Time Slots
a = ( )
CAC(17,3)
b = ( )
c = ( )
d = ( )

8. Image of Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ B
Time Slots
a = ( )
CAC(17,3)
b = ( )
c = ( )
d = ( )

9. Image of Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ B
Time Slots
a = ( )
CAC(17,3)
b = ( )
c = ( )
d = ( )

10. Image of Usage CAC(17,3) M = 4, n = 17, k = 3 Silence Symbol
Survived Packet
Collided Packet
Senders
Receivers A A’ B B’ C C’ D D’
Time Slots
a = ( )
CAC(17,3)
b = ( )
c = ( )
d = ( )

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 Review of the literature of CAC Formulation using Graph Theory
Some new optimal CAC of weight 3 and odd length.

13. Outline Review of the literature of CAC Formulation using Graph Theory
Some new optimal CAC of weight 3 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
Mishima et al., 2009 →
Fu, Lin and Mishima, 2010 →
Jimbo et al., 2007 →

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)

18. We are interested in odd n and k = 3.

19. 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.

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}.
Example (n = 13, k = 3)
x = ( ) ±1 ±2
Δ(x) = {±1, ±2, ±3} = {1, 2, 3, 10, 11, 12}
x = {0, 1, 3} ±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

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 
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)
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}

25. Characterization of Δ Let x be a codeword of a CAC (n, 3).
If Δ(x) = {i}, then i = n/3.
n/3
2n/3

26. 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). i i 2i j i j i 2i

27. 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}, then i + j ≡ ±k (mod n). k i j i
i+j k i j i
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. 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.

30. 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) 1 2 4 3 5
G(11) : 1 2 4 8 3
G(17) : 6 5 7
G(21) : 1 2 4 8 5 10 3 6 9 7

31. Graphical Characterization1 2 4 8 5 10 3 6 9 7 3
Δ = {7} → {0, 7, 14} →
Δ = {1, 2} → {0, 1, 2} →
Δ = {4, 8} → {0, 4, 8} →
Δ ={5, 10} →{0, 5, 10}→
M(21,3) = 5
Δ = {6, 9} →{0, 6, 12}→

32. Strategy
G(n):
even cycles
odd cycles

33. Another Example: CAC(31,3)
{0,4,8}
{0,15,30} 8 15 4
{0,7,14} 1
{0,2,5}
Look for a hyperedge which intersects three distinct odd cycles 2
{0,10,20} 14 3 5 10 7 6
{0,6,12} 13 11
{0,9,18} 12 9
M(31,3) = 7

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

35. Natural Bounds O(n) = number of odd cycles in G(n) Theorem 1
For any odd integer n,

36. More Examples CAC(81, 3) c 9a G(34) : 3b27 9 18 36 3b 3 6 12 24 33 15 30 21 39 1 4 16 17 13 29 35 c 2 8 32 34 26 23 11 40 10 38 31 28 7 22 20 5 19 25 14 37
There is no hyperedges lying across distinct odd cycles.

37. More Examples CAC(81, 3) c M(81,3) = 19 9a G(34) : 3b27 9 18 36 3b 3 6 12 24 33 15 30 21 39 1 4 16 17 13 29 35 c 2 8 32 34 26 23 11 40 10 38 31 28 7 22 20 5 19 25 14 37
There is no hyperedges lying across distinct odd cycles.

38. More Examples CAC(81, 3) c M(81,3) = 19 9a G(34) : 3b27 9 18 36 3b 3 6 12 24 33 15 30 21 39 1 4 16 17 13 29 35 c 2 8 32 34 26 23 11 40 10 38 31 28 7 22 20 5 19 25 14 37
There is no hyperedges lying across distinct odd cycles.

39. Optimal CACs for prime power
Theorem 2
Let p > 3 be a non-Wieferich prime. Then
for r ≥ 1,

40. Optimal CACs for prime power
Theorem 2
Let p > 3 be a non-Wieferich prime. Then
for r ≥ 1,

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.

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

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.
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

45. Thank you for your attention