# Maximal (v, k, 2, 1) Optical Orthogonal Codes with k = 6 and 7 and Small Lengths

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Optical Orthogonal Codes

#### 1.2. Basic Definitions and Notations

**Definition**

**1**

**.**A $(v,k,{\lambda}_{a},{\lambda}_{c})$-OOC can be defined as a collection $\mathcal{C}=\{{C}_{1},\dots ,{C}_{s}\}$ of k-subsets (codeword-sets) of ${Z}_{v}$ such that any two distinct translates of a codeword-set share at most ${\lambda}_{a}$ elements, while any two translates of two distinct codeword-sets share at most ${\lambda}_{c}$ elements:

**Definition**

**2**

**.**The type of C is the number of elements of ${\Delta}^{\prime}C$, i.e., the number of different values of its differences. The type of a codeword is the type of the codeword-set corresponding to it.

**Definition**

**3.**

**Definition**

**5**

**.**Two $(v,k,{\lambda}_{a},{\lambda}_{c})$ optical orthogonal codes $\mathcal{C}$ and ${\mathcal{C}}^{\prime}$ are multiplier equivalent if they can be obtained from one another by an automorphism of ${Z}_{v}$ and the replacement of codeword-sets by some of their translates.

#### 1.3. The Present Paper

## 2. Methods

#### 2.1. The Main Tasks

#### 2.2. Classification Algorithm

#### 2.2.1. Preliminaries

- Lexicographic order:We assume that ${c}_{1}<{c}_{2}<\dots <{c}_{k}$ for each codeword-set $C=\{{c}_{1},{c}_{2},\dots ,{c}_{k}\}$ and define a lexicographic order on the codeword-sets implying that: ${C}^{\prime}=\{{c}_{1}^{\prime},{c}_{2}^{\prime},\dots ,{c}_{k}^{\prime}\}$ is lexicographically smaller than ${C}^{\u2033}=\{{c}_{1}^{\u2033},{c}_{2}^{\u2033},\dots ,{c}_{k}^{\u2033}\}$ if the type of ${C}^{\prime}$ is smaller than that of ${C}^{\u2033}$ or if the types of the two codewords are the same, and ${c}_{i}^{\prime}={c}_{i}^{\u2033}$ for $i<j$ and ${c}_{j}^{\prime}<{c}_{j}^{\u2033}$ for some j.
- Assume ${c}_{1}=0$:If a codeword-set $C\in \mathcal{C}$ is replaced by a translate $C+t\in \mathcal{C}$, an equivalent OOC is obtained. That is why, without loss of generality, we assume that each codeword-set is lexicographically smaller than the codeword-sets of its translates. This means that ${c}_{1}=0$.
- Array of possible codeword-sets:Before the search starts, an array is constructed, which contains all possible codewords, namely all k-subsets of ${Z}_{v}$ that satisfy the auto-correlation property and are smaller than all their translates. They are found in lexicographic order. The automorphisms of ${Z}_{v}$ are applied to each constructed codeword-set. If some automorphism maps the current codeword-set to a smaller set, the current set is not added, because it is already somewhere in the array. If the current set is added to the array, the codeword-sets to which it is mapped by the automorphisms of ${Z}_{v}$ are added right after that, and this makes the tests for the multiplier equivalence of partial solutions very fast.

#### 2.2.2. Exhaustive Backtrack Search

#### 2.3. The Upper Bound ${B}_{1}(v,k,2,1)$

#### 2.4. The Maximum Number of Codewords of a $(v,k,2,1)$-OOC

#### 2.5. Parallel Implementation

## 3. Bound, Maximum Size, and Classification Results

Types of codes by differences | ||

0)3: | 24-2 | |

1)1: | 24-1 | 30-1 |

2)1: | 24-1 | 32-1 |

## 4. Conclusions and Remarks

- The OOCs contain only a few codewords of the three smallest types. For very small lengths, they are an important part of all codewords, but for bigger lengths, they comprise a really small part of all codewords and their effect on the maximal code size becomes almost negligible.
- $M(v,6,2,1)={B}_{0}(v,6,2,1)$ for only six values of $40\le v\le 165$ (40, 42, 44, 45, 60, 74).
- $M(v,7,2,1)<{B}_{0}(v,7,2,1)$ for all $67\le v\le 153$.
- $M(v,6,2,1)={B}_{1}(v,6,2,1)$ for 110 values of $40\le v\le 165$.
- $M(v,7,2,1)={B}_{1}(v,7,2,1)$ for 38 values of $67\le v\le 153$.
- ${B}_{1}(v,6,2,1)<{B}_{0}(v,6,2,1)$ for all $v\ge 91$.
- ${B}_{1}(v,7,2,1)<{B}_{0}(v,7,2,1)$ for all $v\ge 169$.
- The bound we calculated can be approximated in the covered length range with:$${B}_{1}(v,6,2,1)\le \u230a\frac{v}{18}\u230b+f\left(v\right)$$$${B}_{1}(v,7,2,1)=\u230a\frac{v}{24}\u230b.$$

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

OOC | Optical orthogonal codes |

## References

- Chung, F.R.K.; Salehi, J.A.; Wei, V.K. Optical orthogonal codes: Design, analysis, and applications. IEEE Trans. Inf. Theory
**1989**, 35, 595–604. [Google Scholar] [CrossRef] - Moreno, O.; Zhang, Z.; Kumar, P.V.; Zinoviev, V.A. New constructions of optimal cyclically permutable constant weight codes. IEEE Trans. Inf. Theory
**1995**, 41, 448–455. [Google Scholar] [CrossRef] - Yin, J. Some combinatorial constructions for optical orthogonal codes. Discret. Math.
**1998**, 185, 201–219. [Google Scholar] [CrossRef] - Fuji-hara, R.; Miao, Y. Optical orthogonal codes: Their bounds and new optimal constructions. IEEE Trans. Inf. Theory
**2000**, 46, 2396–2406. [Google Scholar] - Chu, W.; Colbourn, C.J. Optimal (n,4,2)-OOC of small order. Discret. Math.
**2004**, 279, 163–172. [Google Scholar] [CrossRef] - Wang, X.; Chang, W. Further results on (v,4,1)-perfect difference families. Discret. Math.
**2010**, 310, 1995–2006. [Google Scholar] [CrossRef] - Baicheva, T.; Topalova, S. Classification of optimal (v,4,1) binary cyclically permutable constant weight codes and cyclic S(2,4,v) designs with v ≤ 76. Probl. Inf. Transm.
**2011**, 47, 224–231. [Google Scholar] [CrossRef] - Chu, W.; Golomb, S.W. A new recursive construction for optical orthogonal codes. IEEE Trans. Inf. Theory
**2003**, 49, 3072–3076. [Google Scholar] - Yang, G.C.; Fuja, T.E. Optical orthogonal codes with unequal auto- and cross-correlation constraints. IEEE Trans. Inf. Theory
**1995**, 41, 96–106. [Google Scholar] [CrossRef] - Buratti, M.; Pasotti, A. Further progress on difference families with block size 4 or 5. Des. Codes Cryptogr.
**2010**, 56, 1–20. [Google Scholar] [CrossRef] - Momihara, K. On cyclic 2(k − 1)-support (n,k)
_{k−1}difference families. Finite Fields Their Appl.**2009**, 15, 415–427. [Google Scholar] [CrossRef] - Dunning, L.A.; Robbins, W.E. Optimal encoding of linear block codes for unequal error protection. Inf. Control
**1978**, 37, 150–177. [Google Scholar] [CrossRef] - Buratti, M.; Momihara, K.; Pasotti, A. New results on optimal (v,4,2,1) optical orthogonal codes. Des. Codes Cryptogr.
**2011**, 58, 89–109. [Google Scholar] [CrossRef] - Buratti, M.; Pasotti, A.; Wu, D. On optimal (v,5,2,1) optical orthogonal codes. Des. Codes Cryptogr.
**2013**, 68, 349–371. [Google Scholar] [CrossRef] - Baicheva, T.; Topalova, S. Optimal (v,4,2,1) optical orthogonal codes with small parameters. J. Comb. Des.
**2012**, 20, 142–160. [Google Scholar] [CrossRef] - Baicheva, T.; Topalova, S. Optimal (v,5,2,1) optical orthogonal codes of small v. Appl. Algebra Eng. Commun. Comput.
**2013**, 24, 165–177. [Google Scholar] [CrossRef] - Momihara, K.; Buratti, M. Bounds and constructions of optimal(n,4,2,1) optical orthogonal codes. IEEE Trans. Inf. Theory
**2009**, 55, 514–523. [Google Scholar] [CrossRef] - Wang, X.; Chang, Y. Further results on optimal (v,4,2,1)-OOCs. Discret. Math.
**2012**, 312, 331–340. [Google Scholar] [CrossRef]

v | ${\mathit{B}}_{0}$ | ${\mathit{B}}_{1}$ | M | OOCs |
---|---|---|---|---|

40 | 2 | 2 | 2 | 1 |

42 | 2 | 2 | 2 | 1 |

44 | 2 | 2 | 2 | 1 |

45 | 2 | 2 | 2 | 2 |

46 | 3 | 2 | 2 | 10 |

47 | 3 | 2 | 2 | 7 |

48 | 3 | 2 | 2 | 58 |

49 | 3 | 2 | 2 | 33 |

50 | 3 | 2 | 2 | 165 |

51 | 3 | 2 | 2 | 200 |

52 | 3 | 2 | 2 | 506 |

53 | 3 | 2 | 2 | 433 |

54 | 3 | 3 | 2 | 2251 |

55 | 3 | 3 | 2 | 1967 |

56 | 3 | 3 | 2 | 6246 |

57 | 3 | 3 | 2 | 6944 |

58 | 3 | 3 | 2 | 15,874 |

59 | 3 | 3 | 2 | 12,861 |

60 | 3 | 3 | 3 | 1 |

61 | 4 | 3 | 3 | 2 |

62 | 4 | 3 | 3 | 9 |

63 | 4 | 3 | 3 | 10 |

64 | 4 | 3 | 3 | 52 |

65 | 4 | 3 | 3 | 42 |

66 | 4 | 3 | 3 | 313 |

67 | 4 | 3 | 3 | 186 |

68 | 4 | 3 | 3 | 987 |

69 | 4 | 3 | 3 | 1250 |

70 | 4 | 3 | 3 | 5654 |

71 | 4 | 3 | 3 | 3477 |

72 | 4 | 4 | 3 | 21,487 |

73 | 4 | 4 | 3 | 13,547 |

74 | 4 | 4 | 4 | 1 |

75 | 4 | 4 | 3 | 91,956 |

76 | 5 | 4 | 3 | 217,428 |

77 | 5 | 4 | 4 | 1 |

78 | 5 | 4 | 4 | 6 |

79 | 5 | 4 | 4 | 10 |

80 | 5 | 4 | 4 | 52 |

81 | 5 | 4 | 4 | 72 |

82 | 5 | 4 | 4 | 428 |

83 | 5 | 4 | 4 | 320 |

84 | 5 | 4 | 4 | 3734 |

85 | 5 | 4 | 4 | 2510 |

86 | 5 | 4 | 4 | 12,360 |

87 | 5 | 4 | 4 | 13,035 |

88 | 5 | 4 | 4 | 65,033 |

89 | 5 | 4 | 4 | 46,355 |

90 | 5 | 5 | 4 | ≥20,925 |

91 | 6 | 5 | 4 | ≥5442 |

92 | 6 | 5 | 4 | ≥26,215 |

93 | 6 | 5 | 5 | 3 |

94 | 6 | 5 | 5 | 12 |

95 | 6 | 5 | 5 | 18 |

96 | 6 | 5 | 5 | 106 |

97 | 6 | 5 | 5 | 95 |

98 | 6 | 5 | 5 | 1150 |

99 | 6 | 5 | 5 | 934 |

100 | 6 | 5 | 5 | ≥1165 |

v | ${\mathit{B}}_{0}$ | ${\mathit{B}}_{1}$ | M | OOCs |
---|---|---|---|---|

67 | 3 | 2 | 2 | 34 |

68 | 3 | 2 | 2 | 108 |

69 | 3 | 2 | 2 | 132 |

70 | 3 | 2 | 2 | 487 |

71 | 3 | 2 | 2 | 384 |

72 | 3 | 3 | 2 | 1497 |

73 | 3 | 3 | 2 | 1208 |

74 | 3 | 3 | 2 | 3735 |

75 | 3 | 3 | 2 | 6087 |

76 | 3 | 3 | 2 | 12,432 |

77 | 3 | 3 | 2 | 13,506 |

78 | 3 | 3 | 2 | 52,070 |

79 | 3 | 3 | 2 | 32,364 |

80 | 3 | 3 | 2 | 132,413 |

81 | 3 | 3 | 2 | 125,433 |

82 | 3 | 3 | 2 | 287,830 |

83 | 3 | 3 | 2 | 240,606 |

84 | 3 | 3 | 2 | 1,279,965 |

85 | 4 | 3 | 3 | 1 |

86 | 4 | 3 | 3 | 1 |

87 | 4 | 3 | 3 | 5 |

88 | 4 | 3 | 3 | 2 |

89 | 4 | 3 | 3 | 8 |

90 | 4 | 3 | 3 | 23 |

91 | 4 | 3 | 3 | 44 |

92 | 4 | 3 | 3 | 84 |

93 | 4 | 3 | 3 | ≥159 |

94 | 4 | 3 | 3 | ≥136 |

v | ${\mathit{B}}_{0}$ | ${\mathit{B}}_{1}$ | M |
---|---|---|---|

101 | 6 | 5 | 5 |

102 | 6 | 5 | 5 |

103 | 6 | 5 | 5 |

104 | 6 | 5 | 5 |

105 | 6 | 5 | 5 |

106 | 7 | 5 | 5 |

107 | 7 | 5 | 5 |

108 | 7 | 6 | 5 |

109 | 7 | 6 | 6 |

110 | 7 | 6 | 6 |

111 | 7 | 6 | 6 |

112 | 7 | 6 | 6 |

113 | 7 | 6 | 6 |

114 | 7 | 6 | 6 |

115 | 7 | 6 | 6 |

116 | 7 | 6 | 6 |

117 | 7 | 6 | 6 |

118 | 7 | 6 | 6 |

119 | 7 | 6 | 6 |

120 | 7 | 6 | 6 |

121 | 8 | 6 | 6 |

122 | 8 | 6 | 6 |

123 | 8 | 6 | 6 |

124 | 8 | 6 | 6 |

125 | 8 | 6 | 6 |

126 | 8 | 7 | 7 |

127 | 8 | 7 | 7 |

128 | 8 | 7 | 7 |

129 | 8 | 7 | 7 |

130 | 8 | 7 | 7 |

131 | 8 | 7 | 7 |

132 | 8 | 7 | 7 |

133 | 8 | 7 | 7 |

134 | 8 | 7 | 7 |

135 | 8 | 7 | 7 |

136 | 9 | 7 | 7 |

137 | 9 | 7 | 7 |

138 | 9 | 7 | 7 |

139 | 9 | 7 | 7 |

140 | 9 | 7 | 7 |

141 | 9 | 7 | 7 |

142 | 9 | 7 | 7 |

143 | 9 | 7 | 7 |

144 | 9 | 8 | 8 |

145 | 9 | 8 | 8 |

146 | 9 | 8 | 8 |

147 | 9 | 8 | 8 |

148 | 9 | 8 | 8 |

149 | 9 | 8 | 8 |

150 | 9 | 8 | 8 |

151 | 10 | 8 | 8 |

152 | 10 | 8 | 8 |

153 | 10 | 8 | 8 |

154 | 10 | 8 | 8 |

155 | 10 | 8 | 8 |

156 | 10 | 8 | 8 |

157 | 10 | 8 | 8 |

158 | 10 | 8 | 8 |

159 | 10 | 8 | 8 |

160 | 10 | 9 | 9 |

161 | 10 | 8 | 8 |

162 | 10 | 9 | 9 |

163 | 10 | 9 | 9 |

164 | 10 | 9 | 9 |

165 | 10 | 9 | 9 |

v | ${\mathit{B}}_{0}$ | ${\mathit{B}}_{1}$ | M |
---|---|---|---|

95 | 4 | 3 | 3 |

96 | 4 | 4 | 3 |

97 | 4 | 4 | 3 |

98 | 4 | 4 | 3 |

99 | 4 | 4 | 3 |

100 | 4 | 4 | 3 |

101 | 4 | 4 | 3 |

102 | 4 | 4 | 3 |

103 | 4 | 4 | 3 |

104 | 4 | 4 | 3 |

105 | 4 | 4 | 3 |

106 | 5 | 4 | 3 |

107 | 5 | 4 | 3 |

108 | 5 | 4 | 3 |

109 | 5 | 4 | 4 |

110 | 5 | 4 | 4 |

111 | 5 | 4 | 4 |

112 | 5 | 4 | 4 |

113 | 5 | 4 | 4 |

114 | 5 | 4 | 4 |

115 | 5 | 4 | 4 |

116 | 5 | 4 | 4 |

117 | 5 | 4 | 4 |

118 | 5 | 4 | 4 |

119 | 5 | 4 | 4 |

120 | 5 | 5 | 4 |

121 | 5 | 5 | 4 |

122 | 5 | 5 | 4 |

123 | 5 | 5 | 4 |

124 | 5 | 5 | 4 |

125 | 5 | 5 | 4 |

126 | 5 | 5 | 4 |

127 | 6 | 5 | 4 |

128 | 6 | 5 | 4 |

129 | 6 | 5 | 4 |

130 | 6 | 5 | 4 |

131 | 6 | 5 | 4 |

132 | 6 | 5 | 4 |

133 | 6 | 5 | 5 |

134 | 6 | 5 | 5 |

135 | 6 | 5 | 5 |

136 | 6 | 5 | 5 |

137 | 6 | 5 | 5 |

138 | 6 | 5 | 5 |

139 | 6 | 5 | 5 |

140 | 6 | 5 | 5 |

141 | 6 | 5 | 5 |

142 | 6 | 5 | 5 |

143 | 6 | 5 | 5 |

144 | 6 | 6 | 5 |

145 | 6 | 6 | 5 |

146 | 6 | 6 | 5 |

147 | 6 | 6 | 5 |

148 | 7 | 6 | 5 |

149 | 7 | 6 | 5 |

150 | 7 | 6 | 5 |

151 | 7 | 6 | 5 |

152 | 7 | 6 | 5 |

153 | 7 | 6 | 5 |

v | ${\mathit{B}}_{0}$ | ${\mathit{B}}_{1}$ |
---|---|---|

31–35 | 2 | 1 |

36–45 | 2 | 2 |

46–53 | 3 | 2 |

54–60 | 3 | 3 |

61–71 | 4 | 3 |

72–75 | 4 | 4 |

76–89 | 5 | 4 |

90–90 | 5 | 5 |

91–105 | 6 | 5 |

106–107 | 7 | 5 |

108–120 | 7 | 6 |

121–125 | 8 | 6 |

126–135 | 8 | 7 |

136–143 | 9 | 7 |

144–150 | 9 | 8 |

151–159 | 10 | 8 |

160–160 | 10 | 9 |

161–161 | 10 | 8 |

162–165 | 10 | 9 |

166–179 | 11 | 9 |

180–180 | 11 | 10 |

181—195 | 12 | 10 |

196–197 | 13 | 10 |

198–210 | 13 | 11 |

211–215 | 14 | 11 |

216–225 | 14 | 12 |

226–233 | 15 | 12 |

234–240 | 15 | 13 |

241–251 | 16 | 13 |

252–255 | 16 | 14 |

256–269 | 17 | 14 |

270–270 | 17 | 15 |

271–285 | 18 | 15 |

286–287 | 19 | 15 |

288–300 | 19 | 16 |

301–303 | 20 | 16 |

304–304 | 20 | 17 |

305–305 | 20 | 16 |

306–315 | 20 | 17 |

316–322 | 21 | 17 |

323–330 | 21 | 18 |

331–339 | 22 | 18 |

340–340 | 22 | 19 |

341–341 | 22 | 18 |

342–345 | 22 | 19 |

346–359 | 23 | 19 |

360–360 | 23 | 20 |

361–375 | 24 | 20 |

376–377 | 25 | 20 |

378–390 | 25 | 21 |

391–395 | 26 | 21 |

396–405 | 26 | 22 |

406–413 | 27 | 22 |

414–420 | 27 | 23 |

421–431 | 28 | 23 |

432–435 | 28 | 24 |

436–449 | 29 | 24 |

450–450 | 29 | 25 |

451–465 | 30 | 25 |

466–467 | 31 | 25 |

468–480 | 31 | 26 |

481–485 | 32 | 26 |

486–495 | 32 | 27 |

496–503 | 33 | 27 |

504–510 | 33 | 28 |

511–519 | 34 | 28 |

520–520 | 34 | 29 |

521–521 | 34 | 28 |

522–525 | 34 | 29 |

526–539 | 35 | 29 |

540–540 | 35 | 30 |

541–555 | 36 | 30 |

556–557 | 37 | 30 |

558–570 | 37 | 31 |

571–575 | 38 | 31 |

576–585 | 38 | 32 |

586–593 | 39 | 32 |

594–600 | 39 | 33 |

601–611 | 40 | 33 |

612–615 | 40 | 34 |

616–629 | 41 | 34 |

630–630 | 41 | 35 |

631–645 | 42 | 35 |

646–646 | 43 | 36 |

647–647 | 43 | 35 |

648–660 | 43 | 36 |

661–664 | 44 | 36 |

665–675 | 44 | 37 |

676–683 | 45 | 37 |

684–690 | 45 | 38 |

691–699 | 46 | 38 |

700–700 | 46 | 39 |

701–701 | 46 | 38 |

702–705 | 46 | 39 |

706–719 | 47 | 39 |

720–720 | 47 | 40 |

v | ${\mathit{B}}_{0}$ | ${\mathit{B}}_{1}$ |
---|---|---|

43–47 | 2 | 1 |

48–63 | 2 | 2 |

64–71 | 3 | 2 |

72–84 | 3 | 3 |

85–95 | 4 | 3 |

96–105 | 4 | 4 |

106–119 | 5 | 4 |

120–126 | 5 | 5 |

127–143 | 6 | 5 |

144–147 | 6 | 6 |

148–167 | 7 | 6 |

168–168 | 7 | 7 |

169–189 | 8 | 7 |

190–191 | 9 | 7 |

192–210 | 9 | 8 |

211–215 | 10 | 8 |

216–231 | 10 | 9 |

232–239 | 11 | 9 |

240–252 | 11 | 10 |

253–263 | 12 | 10 |

264–273 | 12 | 11 |

274–287 | 13 | 11 |

288–294 | 13 | 12 |

295–311 | 14 | 12 |

312–315 | 14 | 13 |

316–335 | 15 | 13 |

336–336 | 15 | 14 |

337–340 | 16 | 14 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Baicheva, T.; Topalova, S. Maximal (*v*, *k*, 2, 1) Optical Orthogonal Codes with *k* = 6 and 7 and Small Lengths. *Mathematics* **2023**, *11*, 2457.
https://doi.org/10.3390/math11112457

**AMA Style**

Baicheva T, Topalova S. Maximal (*v*, *k*, 2, 1) Optical Orthogonal Codes with *k* = 6 and 7 and Small Lengths. *Mathematics*. 2023; 11(11):2457.
https://doi.org/10.3390/math11112457

**Chicago/Turabian Style**

Baicheva, Tsonka, and Svetlana Topalova. 2023. "Maximal (*v*, *k*, 2, 1) Optical Orthogonal Codes with *k* = 6 and 7 and Small Lengths" *Mathematics* 11, no. 11: 2457.
https://doi.org/10.3390/math11112457