Modified group divisible designs with block size 4 and λ = 1

Assaf, Ahmed M.; Wei, Ruizhong

doi:10.1016/s0012-365x(98)00161-7

Cited by 33 publications

(38 citation statements)

References 3 publications

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“…In [18], it is proved that the necessary conditions for the existence of a (3, λ)-MGDD of type t u are also sufficient. From [2,4,9,10,12], we have the following result for (4, λ)-MGDDs of type t u . For the results on (5, 1)-MGDDs of type t u , see [1].…”

Section: Theorem 13 ([16]) There Exists a (3 λ)-Frame Of Type G U mentioning

confidence: 92%

Modified frames with block size 3

Cao¹,

Wu²,

Zhou³

2007

J of Combinatorial Designs

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Section: Theorem 13 ([16]) There Exists a (3 λ)-Frame Of Type G U mentioning

confidence: 92%

Modified frames with block size 3

Cao¹,

Wu²,

Zhou³

2007

J of Combinatorial Designs

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“…For t = 8 and x = 5, by Example 2.7, there exists a (32 × 8, 32, {3, 5}, 1)-CDF, which implies a strictly cyclic {3, 5}-GDD of type 32 8 . Start from this GDD.…”

Section: Cyclic Holey Difference Matricesmentioning

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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“…The subsets fðx i ; y j Þ j 0 j g À 1g(0 i u À 1) are called groups and the subsets fðx i ; y j Þ j 0 i u À 1g(0 j g À 1) are called columns. The existence of a k-MGDD has been solved in [1,2,8,9] when k 4.…”

Section: Introductionmentioning

confidence: 99%