Frame difference families and resolvable balanced incomplete block designs

Costa, Simone; Feng, Tao; Wang, Xiaomiao

doi:10.1007/s10623-018-0472-7

Cited by 19 publications

(24 citation statements)

References 30 publications

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“…The concept of SDF was introduced by Buratti [13] and revisited in [36]. Similar to what has been done in [18,20,21], here we will focus on three particular SDFs with some special “patterns” and we will look for second components: the main ingredients for this purpose will be given by PSs and APSs. More precisely we will use the following three SDFs:

\begin{matrix} true0.33em \\ right ({double-struckZ}_{3}, 4, 4) ‐ SDF : {normalΣ}_{1} & center = & left [[1, 1, - 1, - 1]] . \\ right ({double-struckZ}_{5}, 5, 4) ‐ SDF : {normalΣ}_{2} & center = & left [[0, 1, 1, - 1, - 1]] . \\ right ({double-struckZ}_{45}, 5, 4) ‐ SDF : {normalΣ}_{3} & center = & left [[0, 1, 1, - 1, - 1], [0, 3, 7, 13, 30], [0, 3, 7, 13, 30], [0, 3, 7, 13, 30], \\ right & center & left [0, 3, 7, 13, 30], [0, 5, 14, 26, 34], [0, 5, 14, 26, 34], [0, 5, 14, 26, 34], [0, 5, 14, 26, 34]] . \end{matrix}

…”

Section: Applications To Optical Orthogonal Codesmentioning

confidence: 99%

Partitionable sets, almost partitionable sets, and their applications

Chang

Costa

Feng

et al. 2020

J of Combinatorial Designs

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This paper introduces almost partitionable sets (APSs) to generalize the known concept of partitionable sets. These notions provide a unified frame to construct double-struckZ‐cyclic patterned starter whist tournaments and cyclic balanced sampling plans excluding contiguous units. The existences of partitionable sets and APSs are investigated. As an application, a large number of optical orthogonal codes achieving the Johnson bound or the Johnson bound minus one are constructed.

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\begin{matrix} true0.33em \\ right ({double-struckZ}_{3}, 4, 4) ‐ SDF : {normalΣ}_{1} & center = & left [[1, 1, - 1, - 1]] . \\ right ({double-struckZ}_{5}, 5, 4) ‐ SDF : {normalΣ}_{2} & center = & left [[0, 1, 1, - 1, - 1]] . \\ right ({double-struckZ}_{45}, 5, 4) ‐ SDF : {normalΣ}_{3} & center = & left [[0, 1, 1, - 1, - 1], [0, 3, 7, 13, 30], [0, 3, 7, 13, 30], [0, 3, 7, 13, 30], \\ right & center & left [0, 3, 7, 13, 30], [0, 5, 14, 26, 34], [0, 5, 14, 26, 34], [0, 5, 14, 26, 34], [0, 5, 14, 26, 34]] . \end{matrix}

…”

Section: Applications To Optical Orthogonal Codesmentioning

confidence: 99%

Partitionable sets, almost partitionable sets, and their applications

Chang

Costa

Feng

et al. 2020

J of Combinatorial Designs

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“…The elder constructions are surveyed in [2]. More recent constructions can be found in [8,9,11,14,15,21,22,25]. Here one often tries to have each ∆ g B a complete system of representatives for the cosets of the subgroup of F * q of index µ, namely the group C µ of non-zero µ-th powers of F q .…”

Section: Difference Packings Via Strong Difference Familiesmentioning

confidence: 99%

On silver and golden optical orthogonal codes

Buratti¹

2018

Art Discrete Appl. Math.

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It is several years that no theoretical construction for optimal (v, k, 1) optical orthogonal codes (OOCs) with new parameters has been discovered. In particular, the literature almost completely lacks optimal (v, k, 1)-OOCs with k > 3 which are not regular. In this paper we will show how some elementary difference multisets allow to obtain three new classes of optimal but not regular (3p, 4, 1)-, (5p, 5, 1)-, and (2p, 4, 1)-OOCs which are describable in terms of Pell and Fibonacci numbers. The OOCs of the first two classes (resp. third class) will be called silver (resp. golden) since they exist provided that the square of a silver element (resp. golden element) of Z p is a primitive square of Z p .

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“…Relative difference families have been widely used to construct various kinds of 2‐designs (see [4,10,14] for example). A 2‐

(v, k, λ)

design is a pair

(V, MJX-tex-caligraphicscriptA)

where

V

is a set of

v

points and

MJX-tex-caligraphicscriptA

is a collection of

k

‐subsets of

X

(called blocks ) such that every 2‐subset of

X

is contained in exactly

λ

blocks of

MJX-tex-caligraphicscriptA

.…”

Section: Introductionmentioning

confidence: 99%

“…When

k \in {8, 9}

, it is shown in [1, table 3.3] that a 2‐

(v, k, λ)

design exists if and only if

λ v (v - 1) \equiv 0 (\mod k (k - 1))

and

λ (v - 1) \equiv 0 (\mod k - 1)

with 3 definite exceptions and 160 possible exceptions. Seven of these possible exceptions, that is,

(v, k, λ) \in {(624, 8, 1), (1576, 8, 1), (2025, 9, 1), (765, 9, 2), (1845, 9, 2), (459, 9, 4), (783, 9, 4)}

, were ruled out recently in [13,14] via strong difference families. Following the work in [13], we construct eight more new 2‐

(v, k, λ)

designs in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Existence of strong difference families and constructions for eight new 2‐designs

Wang

Feng

Wang

2021

J of Combinatorial Designs

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Using group rings and characters as in the theory of abelian difference sets, some nonexistence results for strong difference families are provided. Existences of strong difference families with base block size 3 ≤ k ≤ 9 are discussed. Via strong difference families, eight 2‐ ( v , k , λ ) designs whose existences were unknown are constructed for ( v , k , λ ) ∈ { ( 3417 , 8 , 1 ) , ( 3753 , 8 , 1 ) , ( 2665 , 9 , 1 ) , ( 3817 , 9 , 1 ) , ( 4393 , 9 , 1 ) , ( 7353 , 9 , 1 ) , ( 12537 , 9 , 1 ) , ( 345 , 9 , 3 ) }.

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Frame difference families and resolvable balanced incomplete block designs

Cited by 19 publications

References 30 publications

Partitionable sets, almost partitionable sets, and their applications

Partitionable sets, almost partitionable sets, and their applications

On silver and golden optical orthogonal codes

Existence of strong difference families and constructions for eight new 2‐designs

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