Holey Schröder designs of type 3nu1 Wu, Dianhua; Zhang, Hantao
Discrete mathematics,
September 2023, Letnik:
346, Številka:
9
Journal Article
Recenzirano
A holey Schröder design of type h1n1h4n2⋯hknk (HSD(h1n1h4n2⋯hknk)) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h1n1h4n2⋯hknk)) of order n with ni missing subquasigroups (holes) of ...order hi,1≤i≤k, which are disjoint and spanning (i.e., ∑1≤i≤knihi=n). The existence of an HSD(hn) is completely solved, and the existence of HSD(hnu1) for h=1,2, and 4 has been known with a few of possible exceptions. In this paper, we consider the existence of HSD(3nu1). It is shown that for 1≤u≤15 and u≠3, an HSD(3nu1) exists if and only if n(n+2u−1)≡0(mod4), n≥4, and n≥1+2u/3. For 1≤u≤n and u≠3, an HSD(3nu1) exists if and only if n(n+2u−1)≡0(mod4) and n≥4, with possible exceptions of n=29,43. We have also found six new HSDs of type (4nu1).
It is shown that in general, with arbitrary levels of quantitative treatment factor, the constant block-sum balanced incomplete block designs do not exist. I then provide, assuming it exists, a ...general approach to construct a constant block-sum equireplicated block design, possibly with unequally spaced treatments, from the incidence matrix of a block design with same parameters but without constant block-sum property. I illustrate the nuances of obtaining a solution or establishing its nonexistence via several examples of partially balanced incomplete block designs.
Row–column designs are widely recommended for experimental situations when there are two well-identified factors that are cross-classified representing known sources of variability. Row–column ...designs are expected to result a gain in accuracy of estimating treatment comparisons in an experiment as they eliminate the effects of the row and column factors. However, these designs are not readily available when the number of treatments is more than the levels of row and column blocking factors. Here, an algorithmic approach for constructing a new series of row–column designs with incomplete rows and columns, by amalgamating two incomplete block designs has been proposed. A wide range of incomplete block designs, viz., balanced incomplete block designs/ partially balanced incomplete block designs/t-designs, are available in the literature, which can be selected as input designs to construct the proposed series of designs. To avoid the complexity involved in the construction algorithm, an R package “iRoCoDe” has been developed for the generation of the proposed designs. A catalogue of designs has been prepared using “iRoCoDe” for ≤ 20 treatments. Further, a general form of the information matrix of these incomplete row–column designs has been derived, and characterization properties of component designs of the final array have been studied. The designs obtained are cost-effective and efficient as they require less experimental resources and have high canonical efficiency factors.
•Incomplete row-column designs are obtained by fusing two incomplete block designs.•R package is developed to avoid the complexity involved in the construction algorithm.•The developed software is fast and user friendly.•A catalogue of generated IRC designs for v ≤ 20 is prepared for quick reference.
The present article proposes three series of partially balanced incomplete constant block sum designs constructed using regular graphs. Two series develop a three-associate class PBIB designs and one ...develops a five-associate class PBIB design. Moreover, we were able to show the existence of constant block sum PBIB designs with their parameters for a specified association scheme. Towards the end, we have discussed few possible designs, the blocks of which may be constructed from the association schemes discussed in the present research paper.
Two-associate class triangular designs have been explored greatly but the Tm-type PBIB designs (m ≥ 3) largely remains unexplored. The paper is written with an objective to construct a new series of ...Tm-type PBIB designs and to derive some more series of PBIB designs based on these PBIB designs, which we have called as Tm-assisted PBIB designs. For this, we begin by first constructing a series of Tm-type PBIB designs and then based on these designs, three series of Tm-assisted PBIB designs have been constructed. The association schemes of Tm-type and Tm-assisted PBIB designs have been discussed in their complete generalized form.
Resolvable incomplete block designs are often used in many asymmetrical and symmetrical factorial experiments. In this article, two new methods of constructions are proposed to obtain resolvable ...incomplete block designs for asymmetrical and symmetrical factorial experiments. Designs generated using the proposed methods have orthogonal factorial structure and all main effects are estimated with full efficiency and balance. A catalogue of designs obtainable by this method with number of levels of any factor ≤12 is presented along with their efficiency factors.
Circular regular graph designs play an important role in the design of experiments where most of the balanced incomplete block designs require a large number of blocks. In this article, circular ...regular graph designs are constructed in blocks of size four through cyclic shifts. Without studying the complete design, some standard properties of the designs can be observed only through the sets of shifts. Therefore, method of cyclic shifts has an edge over existing methods.
In general, wireless sensors operate with a limited energy source, and energy efficiency is a critical design issue. In order to extend the operation time of wireless sensors, there have been many ...energy efficiency neighbor discovery protocols designed for wireless sensor networks (WSNs) such as Quorum-based, prime-number-based, and block-design-based protocols. Among them, the block-design-based approach was known to be more effective solutions for neighbor discovery in terms of the worst-case discovery latency for a given duty cycle. However, the original block- design-based approach is only applicable to a sensor network where all sensors have the same duty cycle. In order to expand a block-design-based neighbor discovery solution to asymmetric WSNs, we introduce a new neighbor discovery protocol (NDP) that combines two block-design-based schedules to produce a new set of discovery schedules for asymmetric WSNs. Furthermore, by using the Kronecker product method, we prove that any pair of neighboring sensors in the proposed protocol has at least one common active slot within a length of their discovery cycle. Furthermore, the results of the simulation study show that the proposed method is better than representative NDPs (such as Quorum, U-Connect, Disco, SearchLight, Hedis, and Todis) in terms of discovery latency and energy efficiency.
This book presents a systematic, rigorous and comprehensive account of the theory and applications of incomplete block designs. All major aspects of incomplete block designs are considered by ...consolidating vast amounts of material from the literature including the classical incomplete block designs, like the balanced incomplete block (BIB) and partially balanced incomplete block (PBIB) designs. Other developments like efficiency-balanced designs, nested designs, robust designs, C-designs and alpha designs are also discussed, along with more recent developments in incomplete block designs for special types of experiments, like biological assays, test-control experiments and diallel crosses, which are generally not covered in existing books. Results on the optimality aspects of various incomplete block designs are reviewed in a separate chapter, that also includes recent optimality results for test-control comparisons, parallel-line assays and diallel cross experiments.Sample Chapter(s)Chapter 1: Introduction (1,765 KB)Contents: IntroductionAnalysis and Properties of Block DesignsBalanced DesignsPartially Balanced DesignsMore Incomplete Block DesignsOptimality Aspects of Block DesignsReadership: Advanced undergraduates and graduate students, researchers in statistics and applied mathematics.
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