Semi-cyclic holey group divisible designs with block size three

Feng, Tao; Wang, Xiaomiao; Chang, Yanxun

doi:10.1007/s10623-013-9859-7

Cited by 15 publications

(45 citation statements)

References 26 publications

(36 reference statements)

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“…Construction shows that strictly cyclic K ‐GDDs are helpful to yield SCHGDDs. Combining the results of Corollary and Lemma 4.15 in , we have the following strictly cyclic GDDs. Lemma There exists a strictly cyclic

{3, 5}

‐GDD of type 4 t for any

t \geq 4

and

t \notin {5, 8, 11}

.…”

Section: Preliminariesmentioning

confidence: 52%

“…If the length of each block orbit in a K ‐CGDD of type

m^{n}

m n

, then the K ‐GDD is called strictly cyclic . Lemma There exists a strictly cyclic 3‐GDD of type

m^{n}

if and only if

m (n - 1) \equiv 0 (\mod 6)

and

n \geq 4;

n \neg \equiv 2, 3 (\mod 4)

when

m \equiv 2 (\mod 4) .

Construction Suppose that there exist a strictly cyclic K ‐GDD of type

w^{t}

and an l ‐MGDD of type

k^{n}

for each

k \in K

. Then there exists an l ‐SCHGDD of type

(n, w^{t})

.…”

Section: Preliminariesmentioning

confidence: 99%

“…In this paper, we shall focus on the existence of 3‐SCHGDDs of type

(n, m^{t})

with t even. When t is odd, the existence problem has been discussed in recently. Lemma There exists a 3‐SCHGDD of type

(3, m^{t})

if and only if

(t - 1) m \equiv 0 (\mod 2)

and

t \geq 3

with the exception of

m \equiv 0 (\mod 2)

and

t = 3

. Assume that

t \equiv 1 (\mod 2)

and

n \geq 4

.…”

Section: Introductionmentioning

confidence: 99%

“…When t is odd, the existence problem has been discussed in recently. Lemma There exists a 3‐SCHGDD of type

(3, m^{t})

if and only if

(t - 1) m \equiv 0 (\mod 2)

and

t \geq 3

with the exception of

m \equiv 0 (\mod 2)

and

t = 3

. Assume that

t \equiv 1 (\mod 2)

and

n \geq 4

. There exists a 3‐SCHGDD of type

(n, m^{t})

if and only if

t \geq 3

and

(t - 1) n (n - 1) m \equiv 0 (\mod 6)

except when

(n, m, t) = (6, 1, 3)

, and possibly when (1)

n = 6

m \equiv 1, 5 (\mod 6)

and

t \equiv 3, 15 (\mod 18)

, (2)

n = 8

m \equiv 2, 10 (\mod 12)

and

t \equiv 7 (\mod 12)

. Simple counting shows that the number of base blocks in a 3‐SCHGDD of type

(n, m^{t})

(t - 1) n (n - 1) ...

…”

Section: Introductionmentioning

confidence: 99%

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Semi-cyclic Holey Group Divisible Designs and Applications to Sampling Designs and Optical Orthogonal Codes

2014

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Abstract:We consider 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 t is odd and n = 8 or t is doubly even and t = 8, the existence problem is completely solved; when t is singly even, many infinite families are obtained. Applications of our results to twodimensional balanced sampling plans and optimal two-dimensional optical orthogonal codes are also discussed.

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{3, 5}

‐GDD of type 4 t for any

t \geq 4

and

t \notin {5, 8, 11}

.…”

Section: Preliminariesmentioning

confidence: 52%

“…If the length of each block orbit in a K ‐CGDD of type

m^{n}

m n

, then the K ‐GDD is called strictly cyclic . Lemma There exists a strictly cyclic 3‐GDD of type

m^{n}

if and only if

m (n - 1) \equiv 0 (\mod 6)

and

n \geq 4;

n \neg \equiv 2, 3 (\mod 4)

when

m \equiv 2 (\mod 4) .

Construction Suppose that there exist a strictly cyclic K ‐GDD of type

w^{t}

and an l ‐MGDD of type

k^{n}

for each

k \in K

. Then there exists an l ‐SCHGDD of type

(n, w^{t})

.…”

Section: Preliminariesmentioning

confidence: 99%

“…In this paper, we shall focus on the existence of 3‐SCHGDDs of type

(n, m^{t})

with t even. When t is odd, the existence problem has been discussed in recently. Lemma There exists a 3‐SCHGDD of type

(3, m^{t})

if and only if

(t - 1) m \equiv 0 (\mod 2)

and

t \geq 3

with the exception of

m \equiv 0 (\mod 2)

and

t = 3

. Assume that

t \equiv 1 (\mod 2)

and

n \geq 4

.…”

Section: Introductionmentioning

confidence: 99%

“…When t is odd, the existence problem has been discussed in recently. Lemma There exists a 3‐SCHGDD of type

(3, m^{t})

if and only if

(t - 1) m \equiv 0 (\mod 2)

and

t \geq 3

with the exception of

m \equiv 0 (\mod 2)

and

t = 3

. Assume that

t \equiv 1 (\mod 2)

and

n \geq 4

. There exists a 3‐SCHGDD of type

(n, m^{t})

if and only if

t \geq 3

and

(t - 1) n (n - 1) m \equiv 0 (\mod 6)

except when

(n, m, t) = (6, 1, 3)

, and possibly when (1)

n = 6

m \equiv 1, 5 (\mod 6)

and

t \equiv 3, 15 (\mod 18)

, (2)

n = 8

m \equiv 2, 10 (\mod 12)

and

t \equiv 7 (\mod 12)

. Simple counting shows that the number of base blocks in a 3‐SCHGDD of type

(n, m^{t})

(t - 1) n (n - 1) ...

…”

Section: Introductionmentioning

confidence: 99%

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Semi-cyclic Holey Group Divisible Designs and Applications to Sampling Designs and Optical Orthogonal Codes

2014

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“…We give two constructions on k-SCIHGDDs, which can be considered as the generalizations of Constructions 3.1 and 3.4 of [10] respectively.…”

Section: Lemma 318mentioning

confidence: 99%

Determination of Sizes of Optimal Three‐Dimensional Optical Orthogonal Codes of Weight Three with the AM‐OPP Restriction

Wang

Chang

2016

J of Combinatorial Designs

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In this paper, we further investigate the constructions on three-dimensional (u × v × w, k, 1) optical orthogonal codes with the at most one optical pulse per wavelength/time plane restriction (briefly AM-OPP 3-D (u × v × w, k, 1)-OOCs) by way of the corresponding designs. Several new auxiliary designs such as incomplete holey group divisible designs and incomplete group divisible packings are introduced and therefore new constructions are presented. As a consequence, the exact number of codewords of an optimal AM-OPP 3-D (u × v × w, 3, 1)-OOC is finally determined for any positive integers v, w and u ≥ 3.

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Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

Feng

Wang

et al. 2016

J of Combinatorial Designs

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The study of optical orthogonal codes has been motivated by an application in an optical code‐division multiple access system. From a practical point of view, compared to one‐dimensional optical orthogonal codes, two‐dimensional optical orthogonal codes tend to require smaller code length. On the other hand, in some circumstances only with good cross‐correlation one can deal with both synchronization and user identification. These motivate the study of two‐dimensional optical orthogonal codes with better cross‐correlation than auto‐correlation. This paper focuses on optimal two‐dimensional optical orthogonal codes with the auto‐correlation λa and the best cross‐correlation 1. By examining the structures of w‐cyclic group divisible designs and semi‐cyclic incomplete holey group divisible designs, we present new combinatorial constructions for two‐dimensional (n×m,k,λa,1)‐optical orthogonal codes. When k=3 and λa=2, the exact number of codewords of an optimal two‐dimensional (n×m,3,2,1)‐optical orthogonal code is determined for any positive integers n and m≡2(mod4).

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Semi-cyclic holey group divisible designs with block size three

Cited by 15 publications

References 26 publications

Semi-cyclic Holey Group Divisible Designs and Applications to Sampling Designs and Optical Orthogonal Codes

Semi-cyclic Holey Group Divisible Designs and Applications to Sampling Designs and Optical Orthogonal Codes

Determination of Sizes of Optimal Three‐Dimensional Optical Orthogonal Codes of Weight Three with the AM‐OPP Restriction

Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

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