Strong difference families of special types

Chang, Yanxun; Costa, Simone; Feng, Tao; Wang, Xiaomiao

doi:10.1016/j.disc.2019.111776

Cited by 9 publications

(21 citation statements)

References 22 publications

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“…In the following, for each of these values, a 3-pyramidal KTS(n + 3) will be given by means of a (G, {2 3 , 3}, 3, 1)-RDF with G = D, G 1 , D ×V 5 , G 1 ×V 3 and G 2 , respectively. By way of illustration, in the first two cases we follow the instructions of Theorem 3.8 and we concretely construct a 3-pyramidal KTS( 9) and a 3-pyramidal KTS (15).…”

Section: The Smallest Examplesmentioning

confidence: 99%

“…We see that B ∪ {0, a, b} is a system of representatives for the left cosets of J in G 1 , i.e., F is J -resolvable. Following the instructions given in the proof of the "if" part of Theorem 3.8, we obtain the following 3-pyramidal representation of a KTS (15) where, to save space, each element (a, b, c) ∈ G 1 is written as abc. It is known that, up to isomorphism, there exist exactly seven KTS (15), i.e., there are seven non-isomorphic solutions to the well-known Kirkman fifteen schoolgirls problem.…”

Section: A 3-pyramidal Kts(15)mentioning

confidence: 99%

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The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Bonvicini

Buratti

Garonzi

et al. 2021

Des. Codes Cryptogr.

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Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever $$v \equiv 39$$ v ≡ 39 (mod 72), or $$v \equiv 4^e48 + 3$$ v ≡ 4 e 48 + 3 (mod $$4^e96$$ 4 e 96 ) and $$e \ge 0$$ e ≥ 0 , there exists a KTS on v points having at least $$v-3$$ v - 3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism group G acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of G, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.

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Section: The Smallest Examplesmentioning

confidence: 99%

Section: A 3-pyramidal Kts(15)mentioning

confidence: 99%

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Bonvicini

Buratti

Garonzi

et al. 2021

Des. Codes Cryptogr.

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“…Strong difference families have been used to construct OOCs in [15,18]. The idea of strong difference families was also implicitly used in [35,40] to construct OOCs.…”

Section: Applications To Optical Orthogonal Codesmentioning

confidence: 99%

“…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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“…Strong difference families were first introduced explicitly in [5] and systematically developed and discussed in [6–8,11,20] to establish constructions for relative difference families and partitioned difference families. Relative difference families were introduced in [4] as a natural generalization of relative difference sets.…”

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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Strong difference families of special types

Cited by 9 publications

References 22 publications

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Partitionable sets, almost partitionable sets, and their applications

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

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