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

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)

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.

Image of Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ B 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 Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ B 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 Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ B 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 Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ B 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 Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ B 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 Usage CAC(17,3) M = 4, n = 17, k = 3 Senders Receivers A A’ B 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 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 = (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 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 Review of the literature of CAC Formulation using Graph Theory Some new optimal CAC of weight 3 and odd length.

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 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 3 Theorem (Jimbo et al., 2007) Let n = 4t. Then Mishima et al., 2009 → Fu, Lin and Mishima, 2010 → Jimbo et al., 2007 →

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 > 3 Some constructions of optimal CAC of weight 4 and 5  Momihara, Jimbo et al. (2007) For general weight  Kenneth and Wong (2010)

We are interested in odd n and k = 3.

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 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 1 0 1 0 0 0 0 0 0 0 0 0 ) 0 1 2 3 4 5 6 7 8 9 10 11 12 ±1 ±2 Δ(x) = {±1, ±2, ±3} = {1, 2, 3, 10, 11, 12} x = {0, 1, 3} ±3

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.

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

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

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 Δ Let x be a codeword of a CAC (n, 3). If Δ(x) = {i}, then i = n/3. 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. If Δ(x) = {i, j}, then j ≡ ±2i (mod n). i i 2i j i j i 2i

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

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.

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 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

Graphical Characterization 1 2 4 8 5 10 3 6 9 7 3 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 M(21,3) = 5 Δ = {6, 9} →{0, 6, 12}→100000100000100000000

Strategy G(n): even cycles odd cycles

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

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

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

More Examples CAC(81, 3) c 9a G(34) : 3b 27 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.

More Examples CAC(81, 3) c M(81,3) = 19 9a G(34) : 3b 27 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.

More Examples CAC(81, 3) c M(81,3) = 19 9a G(34) : 3b 27 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.

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

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

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

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 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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