Two-Dimensional Optical Orthogonal Codes and Semicyclic Group Divisible Designs

Wang, Jianmin; Yin, Jianxing

doi:10.1109/tit.2010.2043772

Cited by 33 publications

(45 citation statements)

References 30 publications

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“…There is a 3-SCGDD of type m n if and only if n ≥ 3 and Lemma 2.4 [31,32] There is a 4-SCGDD of type m n if and only if n ≥ 4, m(n−1) ≡ 0 (mod 3) and mn(n − 1) ≡ 0 (mod 12), except when n = 4 or (m, n) ∈ {(2, 10), (4, 7), (6, 5)}, and possibly when (1) n = 5, m ≡ ±6 (mod 36) and m ≥ 30, (2) n = 7, m ≡ ±4 (mod 24) and m ≥ 20, (3) n = 10, m ≡ ±2 (mod 12) and m ≥ 10.…”

Section: Lemma 23 [18]mentioning

confidence: 99%

“…Lemma 2.5 [31] Suppose that a k-SCHGDD of type (n, m t ) and a k-SCGDD of type m n exist. Then a k-SCGDD of type (mt) n exists.…”

Section: Lemma 23 [18]mentioning

confidence: 99%

“…We have that if p satisfies p − 5 √ p − 12 > 0, which yields p > 45, then there exists an element x ∈ Z p such that x ∈ C 2 1 , x + 1 ∈ C 2 1 and x − 1 ∈ C 2 0 . When 5 ≤ p ≤ 43, it is readily checked that we may take x as (p, x) = (5, 2), (7,5), (11,6), (13,5), (17,5), (19,2), (23,10), (29,2), (31,11), (37,5), (41, 6), (43,2). ✷ Lemma 6.9 There exists a 3-SCHGDD of type (6, 2 p ) for any prime p ≥ 3.…”

Section: Cyclic Holey Difference Matricesmentioning

confidence: 99%

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Semi-cyclic holey group divisible designs with block size three

Feng

Wang

Chang

2013

Des. Codes Cryptogr.

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In this paper we discuss the existence problem for a semi-cyclic holey group divisible design of type (n, m t ) with block size 3, which is denoted by a 3-SCHGDD of type (n, m t ). When n = 3, a 3-SCHGDD of type (3, m t ) is equivalent to a (3, mt; m)-cyclic holey difference matrix, denoted by a (3, mt; m)-CHDM.It is shown that there is a (3, mt; m)-CHDM if and only if (t − 1)m ≡ 0 (mod 2) and t ≥ 3 with the exception of m ≡ 0 (mod 2) and t = 3. When n ≥ 4, the case of t odd is considered. It is established that if t ≡ 1 (mod 2) and n ≥ 4, then there exists a 3-SCHGDD of type (n, m t ) if and only if t ≥ 3 and (t − 1)n(n − 1)m ≡ 0 (mod 6) with some possible exceptions of n = 6 and 8. The main results in this paper have been used to construct optimal two-dimensional optical orthogonal codes with weight 3 and different auto-and cross-correlation constraints by the authors recently.

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Section: Lemma 23 [18]mentioning

confidence: 99%

“…Lemma 2.5 [31] Suppose that a k-SCHGDD of type (n, m t ) and a k-SCGDD of type m n exist. Then a k-SCGDD of type (mt) n exists.…”

Section: Lemma 23 [18]mentioning

confidence: 99%

Section: Cyclic Holey Difference Matricesmentioning

confidence: 99%

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Semi-cyclic holey group divisible designs with block size three

Feng

Wang

Chang

2013

Des. Codes Cryptogr.

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“…GDDs also have many important applications in coding theory, such as optical orthogonal codes [10,29], constant weight codes [6,9,16], and constant composition codes [6]. The construction of GDDs has became one of the central problems in combinatorial design theory.…”

Section: Introductionmentioning

confidence: 99%

A New Construction of Group Divisible Designs with Nonuniform Group Type

Wei

2016

J. Combin. Designs

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Abstract:The construction of group divisible designs (GDDs) is a basic problem in design theory. While there have been some methods concerning the constructions of uniform GDDs, the construction of nonuniform GDDs remains a challenging problem. In this paper, we present a new approach to the construction of nonuniform GDDs with group type g k m 1 and block size k. We make a progress by proposing a new construction, in which generalized difference sets, a truncating technique, and a difference method are combined to construct nonuniform GDDs. Moreover, we present a variation of this new construction by employing Rees' product constructions. We obtain several infinite families of nonuniform GDDs, as well as many examples whose block sizes are relatively large.

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“…2. k-SCGDDs of type m n are also useful in the construction of optimal one-dimensional optical orthogonal codes (see [31]). Recently, k-SCGDDs of type m n have been used to construct two-dimensional optical orthogonal codes in [28,29].…”

mentioning

confidence: 99%

Semicyclic 4-GDDs and related two-dimensional optical orthogonal codes

Wang

2011

Des. Codes Cryptogr.

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The existence problem for a semicyclic group divisible design (SCGDD) of type m n with block size 4 and index unity, denoted by 4-SCGDD, has been studied for any odd integer m to construct a kind of two-dimensional optical orthogonal codes (2-D OOCs) with the AM-OPPW (at most one-pulse per wavelength) restriction. In this paper, the existence of a 4-SCGDD of type m n is determined for any even integer m, with some possible exceptions. A complete asymptotic existence result for k-SCGDDs of type m n is also obtained for all larger k and odd integer m. All these SCGDDs are used to derive new 2-D OOCs with the AM-OPPW restriction, which are optimal in the sense of their sizes.

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Two-Dimensional Optical Orthogonal Codes and Semicyclic Group Divisible Designs

Cited by 33 publications

References 30 publications

Semi-cyclic holey group divisible designs with block size three

Semi-cyclic holey group divisible designs with block size three

A New Construction of Group Divisible Designs with Nonuniform Group Type

Semicyclic 4-GDDs and related two-dimensional optical orthogonal codes

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