On conflict-avoiding codes of weight 3 and odd length

1
On conflict-avoiding codes ofweight 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.

3
Application to multiple-access collision channel
without feedback
Hello !
message
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

5
Outline

• Review of the literature on CAC
• Formulation using graph theory
• Some new optimal CAC of weight 3 and odd length

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)

for n 4t 2
for odd n, n ??
7
CAC of even length and weight 3

• For n 4t,

• Jimbo, Mishima, Janiszewski, Teymorian
  and Tonchev (07)
• Mishima, Fu and Uruno (09)
• Fu, Lin and Mishima (10)

8
CAC of weight gt 3

• Some constructions of optimal CAC of weight 4 and
  5
• Momihara, Müller, Satoh and Jimbo (07)
• CAC in general
• S and Wong

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

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

11
Equivalent definition of CAC
?(A) x y mod n x, y ? A, x ? y

• The characteristic sets of CAC is a collection of
  subsets of Zn, say A1, A2, , AM , such that
• Each of them has size w.
• ?(Ai) ? ?(Ak) ? for i ? k.
• Example n15,
• 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

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

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.

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A greedy method for equi-difference codewords
n15, m7
A perfect matchingEach vertex is covered by a
red edge,and all red edges are disjoint.
3
4
2
5
6
1
7
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
15
Another example n 31, m 15equi-difference
codewords only
Theorem (Levenshtein (07))The graph with
edgesfrom the equi-difference codewordsare
decomposed into cycles.
Find a maximalmatching
Six equi-differencecodewords
15
8
4
1
2
14
3
10
5
7
6
13
11
12
9
16
The optimal solution with hyperedges
15
8
4
1
2
14
3
10
5
7
6
13
11
12
9
Seven codewords
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M(31,3) 7

• n31
• 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

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The cycle graph for n99.
38
37
23
19
25
31
40
46
34
49
20
7
17
1
10
14
41
2
28
5
29
4
43
47
8
35
26
13
11
32
16
44
22
18
3
9
6
36
48
27
45
24
12
15
42
30
33
39
21
19
M(99,3) 24

• Two non-equi-difference codewords 0,1,11,
  0,6,15.
• Twenty two equi-difference codewords generated by
  2,7,8,12,13,17,18,19,20,21,22,23,25,27,28,29,30,3
  1,32,33,47,48.

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A sufficient condition for being an optimal CAC

• Theorem 1
• Let n be an odd integer, and let Nodd(n) be the
  number of odd-cycle in the graph.
• If we can find ?Nodd(n) / 3? mutually disjoint
  hyperedge of size 3 lying across 3? ?Nodd(n) / 3?
  cycles of odd length, then equality holds.

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The upper bound in Thm 1 is not tight

• Theorem 2 for e ?1,

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.)
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n 11 13 15 17 19 21 23 25 27 29
M(n,3) 2 3 4 4 4 5 5 6 6 7
Thm 2
n 31 33 35 37 39 41 43 45 47 49
M(n,3) 7 8 8 9 10 10 10 11 11 11
new new new Thm 2
n 51 53 55 57 59 61 63 65 67 69
M(n,3) 13 13 13 14 14 15 15 16 16 17
new
n 71 73 75 77 79 81 83 85 87 89
M(n,3) 17 17 19 18 19 19 20 21 22 21
new Thm 2 new
n 91 93 95 97 99 101 103 105 107 109
M(n,3) 22 23 23 24 24 25 25 26 26 27
new new
non-equiv-difference codewords are required to
construct optimal CAC
23
Conclusion

• Numerical results
• For all odd n lt500, except n81, 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

8134, 189 33 ? 7 24335, 34373, 405 34 ?
5, 441 32 ? 72.
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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 n4m 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.

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n 111 113 115 117 119 121 123 125 127 129
M(n,3) 28 28 28 29 29 29 31 31 30 31
Thm 1 Thm 1
n 131 133 135 137 139 141 143 145 147 149
M(n,3) 32 32 33 34 34 35 35 36 36 37

n 151 153 155 157 159 161 163 165 167 169
M(n,3) 36 38 38 39 40 39 40 41 41 42
Thm 1 Thm 1 Thm 1 Thm 1
n 171 173 175 177 179 181 183 185 187 189
M(n,3) 41 43 43 44 44 45 46 46 46 47
Thm 1 Thm 1 Similar to Thm 2
n 191 193 195 197 199 201 203 205 207 209
M(n,3) 47 48 49 49 49 50 50 51 51 51
Thm 1 Thm 1
non-equiv-difference codewords are required to
construct optimal CAC
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n 211 213 215 217 219 221 223 225 227 229
M(n,3) 52 53 53 52 54 55 55 56 56 57
Thm 1 Thm 1 Thm 1 Thm 1
n 231 233 235 237 239 241 243 245 247 249
M(n,3) 57 57 58 59 59 60 60 60 61 62
Thm 1 Thm 1 Thm 2 Thm 1
n 251 253 255 257 259 261 263 265 267 269
M(n,3) 60 62 64 64 64 65 65 66 65 67
Thm 1 Thm 1
n 271 273 275 277 279 281 283 285 287 289
M(n,3) 67 68 68 69 69 69 70 71 71 72
Thm 1 Thm 1 Thm 1 Thm 1
n 291 293 295 297 299 301 303 305 307 309
M(n,3) 73 73 73 73 74 74 76 76 76 77
Thm 1 Thm 1 Thm 1
non-equiv-difference codewords are required to
construct optimal CAC
27
n 311 313 315 317 319 321 323 325 327 329
M(n,3) 77 78 78 79 79 80 80 81 82 81
Thm 1 Thm 1
n 331 333 335 337 339 341 343 345 347 349
M(n,3) 80 83 83 82 85 84 85 86 86 87
Thm 1 Thm 1 Thm 1 Thm 2
n 351 353 355 357 359 361 363 365 367 369
M(n,3) 87 88 88 89 89 89 88 89 91 92
Thm 1 Thm 1
n 371 373 375 377 379 381 383 385 387 389
M(n,3) 92 93 94 94 94 94 95 95 94 97
Thm 1 Thm 1
n 391 393 395 397 399 401 403 405 407 409
M(n,3) 97 98 98 99 99 100 100 101 101 102
Thm 1 Thm 1 Thm 1 Similar to Thm 2
non-equiv-difference codewords are required to
construct optimal CAC
28
n 411 413 415 417 419 421 423 425 427 429
M(n,3) 103 102 103 104 104 105 105 106 106 107
Thm 1 Thm 1
n 431 433 435 437 439 441 443 445 447 449
M(n,3) 106 108 109 108 109 110 110 110 112 112
Thm 1 Thm 1 Similar to Thm 2 Thm 1
n 451 453 455 457 459 461 463 465 467 469
M(n,3) 112 112 113 114 114 115 115 116 116 116
Thm 1 Thm 1
n 471 473 475 477 479 481 483 485 487 489
M(n,3) 118 116 118 119 119 120 120 121 121 122
Thm 1 Thm 1 Thm 1
n 491 493 495 497 499 501 503 505 507 509
M(n,3) 122 123 123 123 124 125 125 126 127 127
Thm 1 Thm 1 Thm 1
non-equiv-difference codewords are required to
construct optimal CAC
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n 511 513 515 517 519 521 523 525 527 529
M(n,3) 103 102 103 104 104 105 105 106 106 107

n 531 533 535 537 539 541 543 545 547 549
M(n,3) 106 108 109 108 109 110 110 110 112 112

n 451 453 455 457 459 461 463 465 467 469
M(n,3) 112 112 113 114 114 115 115 116 116 116

n 471 473 475 477 479 481 483 485 487 489
M(n,3) 118 116 118 119 119 120 120 121 121 122

n 491 493 495 497 499 501 503 505 507 509
M(n,3) 122 123 123 123 124 125 125 126 127 127

non-equiv-difference codewords