Optimal Conflict-avoiding Codes of Odd Length Weight Three

1. 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

2. (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.

3. 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.

4. 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)

5. 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)

6. 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)

7. 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)

8. 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)

9. 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)

10. 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)

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. 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

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. ±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}

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 ∈xfor 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 forx≠ y

23. 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

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. 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.

26. 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).

27. 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).

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

31. 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

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

33. 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

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) 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.

37. 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.

38. 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.

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. 2011.

45. Thank you for your attention