1. On conflict-avoiding codes of weight 3 and odd length Kenneth Shum Oct 2011

2. Definitions Optical orthogonal code OOC(n,w, a, c ) – Length n – Weight w – Hamming auto-correlation  a – Hamming cross-correlation  c Conflict-avoiding code (Tsybakov and Rubinov (02)) CAC(n,w) = OOC(n,w, ,1) no requirement on Hamming auto-correlation. Oct 2011kshum 2

3. Application to multiple-access collision channel without feedback Oct 2011kshum 3 Hello ! message Scheduler Packet 1Packet 2Packet 3Packet 4 Channel coding (Reed-Solomon) Any two packets can recover the original message x Receiver x x Other user Hello ! collision

4. Parameters Number of codewords = T – Total number of potential users – Each user is statically assigned a unique codeword Sequence period = n – maximal delay experience by an active user Hamming weight = w – Maximal number of simultaneously active users Objective: Given n and w, maximize T Oct 2011kshum 4

5. Outline Review of the literature on CAC Formulation using graph theory Some new optimal CAC of weight 3 and odd length Oct 2011kshum 5

6. Maximal number of codewords Let M(n,w) be largest number of codewords in a CAC of length n and weight w. Levenshtein (07) Oct 2011kshum 6 for n = 4t + 2 for odd n, n 

7. CAC of even length and weight 3 For n = 4t, Oct 2011kshum 7 Jimbo, Mishima, Janiszewski, Teymorian and Tonchev (07) Mishima, Fu and Uruno (09) Fu, Lin and Mishima (10)

8. CAC of weight > 3 Some constructions of optimal CAC of weight 4 and 5 – Momihara, Müller, Satoh and Jimbo (07) CAC in general – S and Wong Oct 2011kshum 8 For w  3,

9. Outline Review of the literature on CAC Formulation using graph theory – hypergraph matching Some new optimal CAC of weight 3 and odd length Oct 2011kshum 9

10. Terminology A binary sequence can be represented by a characteristic set. – Sequence: 0 1 1 0 0 1 0 0  {1,2,5} Indices 0 1 2 3 4 5 6 7 The set of differences contains the separations between the ones in a sequence –  (A) = {x – y mod n: x, y  A, x  y} – For example  ({1,2,5}) = {1,3,4,5,7} Oct 2011kshum 10 0 1 1 0 0 1 0 0

11. Equivalent definition of CAC The characteristic sets of CAC is a collection of subsets of Z n, say A 1, A 2, …, A M, such that – Each of them has size w. –  (A i )   (A k ) =  for i  k. Example: n=15, – 111000000000000  {0,1,2},  ({0,1,2}) = {1,2,13,14} – 100100100000000  {0,3,6},  ({0,3,6}) = {3,6,9,12} – 100010001000000  {0,4,8},  ({0,4,8}) = {4,7,8,11} – 100001000010000  {0,5,10},  ({0,5,10}) = {5,10} Oct 2011kshum 11 distinct  (A) = {x – y mod n: x, y  A, x  y}

12. Equi-difference codewords By cyclically shifting the sequence, we can assume without loss of generality that 0 belongs to the characteristic set. For sequence with Hamming weight 3, we can write the characteristic set as {0,a,b} WLOG. –  ({0,a,b}) = {  a,  b,  (a – b)} In particular, a sequence with characteristic set {0,a,2a} is said to be equi-difference. – The integer a is called the generator of this codeword –  ({0,a,2a}) = {  a,  2a} Oct 2011kshum 12 100010000010000000 a b-a b 100010001000000000 a a 2a

13. Formulation using (hyper)graph Observation: x   (A) implies n – x   (A)  we can identify x and –x mod n. Assume n odd. Let m = (n – 1)/2. Undirected graph with vertex set {1,2,…,m}. Construct hyperedges  ({0,a,b})  {1,2,…,m} – for a and b running over all distinct elements in {1,2,…,n} Objective: look for a maximal collection of non- intersecting hyperedges. – A matching problem. Oct 2011kshum 13

14. A greedy method for equi- difference codewords Oct 2011kshum 14 1 3 5 n=15, m=7 2 4 7 6 111000000000000  {0,1,2}   ({0,1,2}) = {  1,  2}. 101010000000000  {0,2,4}   ({0,2,4}) = {  2,  4}. (conflict with {0,1,2}) 100100100000000  {0,3,6}   ({0,3,6}) = {  3,  6}. 100010001000000  {0,4,8}   ({0,4,8}) = {  4,  7}. 100001000010000  {0,5,10}   ({0,5,10}) = {  5}. 100000100000100  {0,6,12}   ({0,6,12}) = {  3,  6} (conflict with {0,3,6}) 100000010000001  {0,7,14}   ({0,7,14}) = {  1,  4} (conflict with {0,1,2}) and {0,4,8} A perfect matching Each vertex is covered by a red edge, and all red edges are disjoint.

15. Another example: n = 31, m = 15 equi-difference codewords only Oct 2011kshum 15 8 4 1 12 2 14 3 5 10 13 11 7 6 9 {0,1,2} {0,4,8} {0,5,10} {0,9,18} {0,3,6} {0,7,14} Theorem (Levenshtein (07)) The graph with edges from the equi-difference codewords are decomposed into cycles. Find a maximal matching Six equi-difference codewords

16. The optimal solution with hyperedges Oct 2011kshum 16 8 15 4 1 12 2 14 3 5 10 13 11 7 6 9 {0,15,30} {0,4,8} {0,10,20} {0,9,18} {0,6,12} {0,7,14} Seven codewords {0,6,13} An example of hyperedge {0,2,6} Another example of hyperedge {0,2,5} Look for a hyperedge which intersects three distinct odd cycles

17. M(31,3) = 7 n=31 – {0,4,8}  1000100010000000000000000000000 – {0,6,12}  1000001000001000000000000000000 – {0,7,14}  1000000100000010000000000000000 – {0,9,18}  1000000001000000001000000000000 – {0,10,20}  1000000000100000000010000000000 – {0,15,30}  1000000000000001000000000000001 – {0,2,5}  1010010000000000000000000000000 Oct 2011kshum 17

18. The cycle graph for n=99. Oct 2011kshum 18 1 2 4 8 16 32 35 29 41 17 34 49 25 37 31 7 14 28 43 13 26 47 5 10 20 40 46 23 38 19 11 22 44 48 24 9 12 6 3 18 36 2745 42 15 30 3921 33 {0,1,11} {0,6,15}

19. M(99,3) = 24 Two non-equi-difference codewords: {0,1,11}, {0,6,15}. Twenty two equi-difference codewords generated by 2,7,8,12,13,17,18,19,20,21,22,23,25, 27,28,29,30,31,32,33,47,48. Oct 2011kshum 19

20. A sufficient condition for being an optimal CAC Theorem 1: – Let n be an odd integer, and let N odd (n) be the number of odd-cycle in the graph. – If we can find  N odd (n) / 3  mutually disjoint hyperedge of size 3 lying across 3   N odd (n) / 3  cycles of odd length, then equality holds. Oct 2011kshum 20

21. The upper bound in Thm 1 is not tight Theorem 2: for e  1, Oct 2011kshum 21 For n= powers of 3 or 7, M(n,3) is strictly less than the upper bound in Theorem 1. (because in these cases, non-equi-difference codewords are not useful in constructing optimal CAC.)

22. Oct 2011kshum 22 n11131517192123252729 M(n,3)2344455667 Thm 2 n31333537394143454749 M(n,3)7*7*8*8*8910 10*11 new Thm 2 n51535557596163656769 M(n,3)13 14*1415 16 17 new n71737577798183858789 M(n,3)1717*191819 20212221* newThm 2new n9193959799101103105107109 M(n,3)2223*232424*25 26 27 new * non-equiv-difference codewords are required to construct optimal CAC

23. Conclusion Numerical results: – For all odd n <500, except n=81, 189, 243, 343, 405, 441, – M(81,3) = 19, M(189,3) = 47 – M(243,3) = 60, M(343,3) = 85 – M(405,3) = 101, M(441,3) = 110 Oct 2011kshum 23 81=3 4, 189 = 3 3  7 243=3 5, 343=7 3, 405 = 3 4  5, 441 = 3 2  7 2.

24. References Tsybakov and Rubinov, Some constructions of conflict-avoiding codes, Problems of Inf. Trans., 2002. V. I. Levenshtein, Conflict-avoiding codes and cyclic triple systems, Probems of Inf. Trans., 2007. M. Jimbo et al., On conflict-avoiding codes of length n=4m for three active users, IEEE Trans. Inf. Theory, 2007. M. Mishima, H.-L. Fu and S. Uruno, Optimal conflict-avoiding codes of length n  0(mod16) and weight 3, Des. Codes Cryptogr., 2009. H.-L. Fu, Y.-H. Lin and M. Mishima, Optimal conflict-avoiding codes of even length and weight 3, IEEE Trans. Inf. Theory, 2010. K. W. Shum and W. S. Wong, A tight asymptotic bound on the size of constant-weight conflict-avoiding codes, Des. Codes Cryptogr., 2010. Oct 2011kshum 24

25. Oct 2011kshum 25 n111113115117119121123125127129 M(n,3)28 29 31 30*31* Thm 1 n131133135137139141143145147149 M(n,3)32 3334 35 36 37 n151153155157159161163165167169 M(n,3)36*3838*394039*4041*4142 Thm 1 n171173175177179181183185187189 M(n,3)41*43 44*444546 47 Thm 1 Similar to Thm 2 n191193195197199201203205207209 M(n,3)474849 50*5051 51* Thm 1 * non-equiv-difference codewords are required to construct optimal CAC

26. Oct 2011kshum 26 n211213215217219221223225227229 M(n,3)525353*52*54*5555*56 57 Thm 1 n231233235237239241243245247249 M(n,3)57* 5859 60 6162* Thm 1 Thm 2Thm 1 n251253255257259261263265267269 M(n,3)60*6264 65 6665*67 Thm 1 n271273275277279281283285287289 M(n,3)6768 6969* 70*71*7172 Thm 1 n291293295297299301303305307309 M(n,3)73 73*7474*76 76*77 Thm 1 * non-equiv-difference codewords are required to construct optimal CAC

27. Oct 2011kshum 27 n311313315317319321323325327329 M(n,3)7778 79 80*80818281* Thm 1 n331333335337339341343345347349 M(n,3)80*83 82*8584*8586 87 Thm 1 Thm 2 n351353355357359361363365367369 M(n,3)8788 89 88*89*9192 Thm 1 n371373375377379381383385387389 M(n,3)929394 94*95 94*97 Thm 1 n391393395397399401403405407409 M(n,3)9798*989999*100100*101 102 Thm 1 Similar to Thm 2 * non-equiv-difference codewords are required to construct optimal CAC

28. Oct 2011kshum 28 n411413415417419421423425427429 M(n,3)103102103104*104105 106 107* Thm 1 n431433435437439441443445447449 M(n,3)106*108109108109*110 110*112 Thm 1 Similar to Thm 2 Thm 1 n451453455457459461463465467469 M(n,3)112112*113114 115 116*116 Thm 1 n471473475477479481483485487489 M(n,3)118116*118119 120120*121 122* Thm 1 n491493495497499501503505507509 M(n,3)122123123* 124*125 126127 Thm 1 * non-equiv-difference codewords are required to construct optimal CAC

29. Oct 2011kshum 29 n511513515517519521523525527529 M(n,3)103102103104*104105 106 107* n531533535537539541543545547549 M(n,3)106*108109108109*110 110*112 n451453455457459461463465467469 M(n,3)112112*113114 115 116*116 n471473475477479481483485487489 M(n,3)118116*118119 120120*121 122* n491493495497499501503505507509 M(n,3)122123123* 124*125 126127 * non-equiv-difference codewords