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Konstrukcija blokovnih dizajna s određenim dopustivim parametrima se često pokušava izvesti za određeni skup parametara uz pretpostavljanje nekih dodatnih ograničenja na strukturu dizajna kako bi pretraživanje bilo računalno izvedivo. Prirodno ograničenje je pretpostavka da određena grupa automorfizama djeluje na dizajn. Jedna od metoda konstruiranja blokovnih dizajna s pretpostavljenom grupom automorfizama je metoda koja koristi orbitne matrice, a sastoji se od dvaju koraka: konstrukcije orbitnih matrica za pretpostavljenu grupu automorfizma i konstrukcije blokovnih dizajna za orbitne matrice dobivene na ovaj način (ovaj korak se naziva indeksiranje orbitnih matrica). Indeksiranje orbitnih matrica obično se provodi metodom iscrpnog pretraživanja. Međutim, ponekad iscrpno pretraživanje nije izvedivo jer postoji previše slučajeva koje bi trebalo provjeriti. Slično, jako regularni grafovi s određenim dopustivim parametrima se mogu konstruirati korištenjem orbitnih matrica za pretpostavljenu grupu automorfizma. Genetski algoritam je metoda pretraživanja koja se koristi u računarstvu za pronalazak točnih ili približno točnih rješenja za probleme optimizacije i pretraživanja. Genetski algoritam oponaša prirodnu evoluciju, odnosno temelji se na optimizaciji populacije, podskupa cijelog prostora pretraživanja. Kao i u prirodi, populacija se sastoji od jedinki koje se mogu razmnožavati i nad kojima mogu djelovati određene mutacije te se na taj način stvaraju nove jedinke s boljim ili lošijim svojstvima od prethodnih. Cilj algoritma je usmjeriti populaciju ka stvaranju boljih jedinki što može rezultirati pronalaskom optimalnih rješenja zadanog problema. U ovoj doktorskoj disertaciji opisujemo korištenje genetskog algoritma u koraku indeksiranja orbitnih matrica za konstrukciju blokovnih dizajna i jako regularnih grafova s pretpostavljenom grupom automorfizama. Korištenjem ovog pristupa konstruirali smo nove blokovne dizajne, točnije nove Steinerove sustave s parametrima S(2,5,45) i nove simetrične dizajne s parametrima (71,15,3) te nove jako regularne grafove s parametrima (96,19,2,4) i (96,20,4,4). Construction of block designs with certain admissible parameters is often attempted for a particular set of parameters with the assumption of some additional constraints on the design structure in order to make the search computationally feasible. A natural constraint is the assumption that a given group of automorphisms acts on the design. One of the methods for constructing block designs with a prescribed automorphism group is the method that uses so called orbit matrices. It consists of two steps: construction of orbit matrices for the given automorphism group and construction of block designs for the orbit matrices obtained in this way (this step is called "indexing of orbit matrices"). Indexing is usually performed by exhaustive search. However, sometimes exhaustive search is not feasible because there are too many cases to check. Similarly, strongly regular graphs with certain admissible parameters can be constructed using orbit matrices for the prescribed automorphism group. Genetic algorithms are search methods used in computing whose objective is to find exact or approximate solutions to optimization and search problems. A genetic algorithm mimics natural evolution, that is, it is based on optimizing a population (a subset of the entire search space). As in nature, the population consists of individuals that can reproduce and that can be affected by certain mutations, thus creating new individuals with better or worse properties than the previous ones. The goal of the algorithm is to direct the population towards creating better individuals, which can result in finding optimal solutions to a given problem. In this doctoral dissertation, we describe the use of a genetic algorithm in the step of indexing of orbit matrices for the construction of block designs and strongly regular graphs with a prescribed automorphism group. Using this approach, we have constructed new block designs, namely new Steiner systems with parameters S(2,5,45), new symmetric designs with parameters (71,15,3) and new strongly regular graphs with parameters (96,19,2,4) and (96,20,4,4).

In agricultural, animal, fisheries and industrial experimentation under block design setup, systematic trend may affect the response under consideration. Although remote, these effects may still ha...

Controlled experiments give researchers a statistical tool for determining the yield from subjecting an experimental unit to various treatments. We will discuss a replicated, block design applied to the experimental unit yeast. We subjected the yeast to six treatments. The purpose of the experiment is to extract a compound to be used in the manufacturing industry. We considered an ANOVA and a MANOVA model to analyze the data. The rationale for selecting one model over the other will be discussed. Results and recommendations of which treatments to use when processing the yeast will be presented, also.

Presence of one or more outliers in the observations affect inference procedure in statistical analysis. Designs robust against presence of a single outlier can be found in the literature for both ...

Balanced incomplete block design (BIBD) offers an important way to construct non-homogeneous quantum low-density-parity-check (LDPC) codes of Calderbank-Shor-Steane (CSS) type. However, the rates of the quantum codes based on this method are limited to be $1-2/m$ approximately, where $m$ , the number of base blocks chosen from the associated BIBD, is even. We overcome this limitation by providing a rate scalable way to construct non-homogeneous quantum LDPC codes of CSS type. With our method, the rates of the resultant quantum codes are $1-2/m'$ , where $m'=m/e$ and $e\in \{1, 2\}$ . Note that here $m'$ can be odd . Thus, we double the rate range of the codes yielded from the existing strategies. Moreover, one can switch codes between the two rates flexibly adapting to various application scenarios.

In block design, construction of Balanced Incomplete Block Design (BIBD) remained an unsolved problem in combinatorial design; also various construction techniques have been introduced to build the elements of BIBDs for specific parameters; no general method has been proposed to find a suitable structure for BIBDs. This paper aim at employing Finite Euclidean Geometry FEG (N,s) of N – dimensional space to construct balanced incomplete block design (BIBD). Also geometrical construction of FEG (2,2) BIBDs has been made. The results show that this technique proved a better method for constructing BIBD than other methods in terms of estimation of parameters to build the design structure.

In agricultural experiments, if outliers are present in a data set, inference of experiment may be reversed. The purpose of this article is to develop a method for detection of subset of outlier ve...

In this article, we present a gauge R & R study that is different from the standard setup in several ways. A balanced incomplete block design (BIBD) was used to collect the data because the measure...

In this paper, we identify and characterize a family of sampling designs such that, under these designs, the sample median is a median-unbiased estimator of the population median. We first consider the simple random sampling case. A simple random sampling design has the median-unbiasedness property. Moreover, upon deleting samples from the simple random sampling case and imposing a uniform probability distribution on the remaining samples, the sample median is a median-unbiased estimator provided that the support meets a minimum threshold. However, there are other sampling designs, such as those based on balanced incomplete block designs, that do not need to meet the minimum threshold requirement to have the sample median be a median-unbiased estimator. We construct non-uniformly distributed sampling designs that have the median-unbiasedness property as well. In fact, the sample median is a best linear unbiased estimator within the class of linear median unbiased estimators. We show the sample median follows the Gauss-Markov Property under a simple random sampling design.

In this paper we have proved that for a putative symmetric block design D with parameters (210,77,28), admitting a Frobenius group G = (hρ, μ|ρ19 = μ3 = 1, ρμ = ρ7) of order 57, there are exactly six possible orbit structures up to isomorphism with the orbit distribution [1; 19; 19; 19; 19; 19; 19; 19; 19; 19; 19; 19].

We prove that any symmetric block design (v, k, λ) generates optimal ternary and quaternary constant-weight equidistant codes, whose parameters n, N, w, d, q are uniquely determined by the parameters of the block design. For one rather special case, we construct symbolwise uniform equidistant codes of the minimum length.

When data for a completely randomised design are categorical rather than continuous, the Cochran–Mantel–Haenszel suite of tests may be applied, albeit that there is only one stratum. In introductory design courses this design, and the randomised block design, are usually the first designs introduced. Using the completely randomised design instead of the randomised block design ignores blocks (strata) and results in expressions for the tests statistics that are simpler than if the usual general test statistics are applied. Based on models given here, the usual Cochran–Mantel–Haenszel tests for the completely randomised design are shown to be Wald-type tests.

A pair of orthogonal latin cubes of order $q$ is equivalent to an MDS code with distance $3$ or to an ${\rm OA}_1(3,5,q)$ orthogonal array. We construct pairs of orthogonal latin cubes for a sequence of previously unknown orders $q_i=16(18i-1)+4$ and $q'_i=16(18i+5)+4$. The minimal new obtained parameters of orthogonal arrays are ${\rm OA}_1(3,5,84)$. Keywords: latin square, latin cube, MOLS, MDS code, block design, Steiner system, orthogonal array, Comment: New pairs of orthogonal latin 3-cubes are available on the website https://ieee-dataport.org/open-access/graeco-latin-cubes

The concept of a locally repairable code (LRC) was introduced to protect the data from disk failures in large-scale storage systems. In this paper, we consider the LRCs with multiple disjoint repair sets and each repair set contains exactly one check symbol. By using several structures from combinatorial design theory, such as balanced incomplete block design, cyclic packing, group divisible design, near-Skolem sequence and Langford sequence, we construct several infinite classes of LRCs with the size of each repair set at most 3, which are optimal with respect to the bound proposed by Rawat et al. in 2016.

Wheat breeders frequently evaluate large numbers of germplasm and segregating populations for selection in field trials. For testing large number of germplasms under complete block design not much suitable because their assumption to accommodate all the treatment combination in each block that increase the size of experiment and reduce the accuracy. To avoid this constraint the present experiment was designed with objective to realise the suitability of alpha lattice design (ALD) over randomised complete block design (RCBD) in plant breeding experiments. The present experiment was conducted in ALD of 8 × 20 i.e. eight blocks each with 20 genotypes in two replications under rainfed and restricted irrigated conditions to estimate the drought tolerance indices. Nine drought tolerance indices were estimated using the data of rainfed and restricted irrigated conditions. These drought tolerance indices were subjected to analysis of variances of both the designs ALD and RCBD to know the efficiency of ALD over RCBD. The relative efficiency of trials shows that ALD was more efficient than RCB design. Based on grain yield per plant under both the conditions twenty-one tolerant genotypes and eleven susceptible genotypes were identified. Identified tolerant and susceptible genotypes can be used for development of mapping populations and for QTLs mapping programme for drought tolerance in wheat (Triticum aestivum L.).

An incomplete-block design defines both a concurrence graph and a Levi graph. Properties of either graph can be used to compare designs with respect to D-optimality and with respect to A-optimality. In this paper, we show that optimality of the design implies strong conditions on connectivity properties of the graph, and use this to classify the optimal designs when the number of observational units is close to minimal.

In a granulometric analysis of coffee beans with different categories of defects, the data can be organized in contingency tables, and when considering the discrimination by harvest, they may have a structure that suggest a more complex model, by means of the counting of defective coffee beans compared to different crops interacting with the classification of defects and percentages of sieve grains, which characterizes a block design with multivariate responses. However, due to the techniques based on the analysis of variance, considering the uniform correlation structure for all plots, it becomes feasible to propose a model that allows contemplating different structures between the plots, associating the effects of the crops to the defects in the granulometric procedure applied to the coffee beans. Thus, the hypothesis of incorporating the effects of crops associated with defects arises using the biplot multivariate technique. This work aims to propose the use of corrected biplots by predictions obtained trhough the fit to the Generalized Linear Model in the coffee grain size classification, broken down by components of the effect of the harvests. In conclusion, the use of GEE models with the corrected biplot technique by the predictions is feasible for application to be applied to the granulometric analysis of defective coffee beans, presenting discrimination regarding the effects of harvests.

Structured LDPC codes have been constructed using balanced incomplete block (BIB) designs, resolvable BIB designs, mutually orthogonal Latin rectangles, partial geometries, group divisible designs, resolvable group divisible designs and finite geometries. Here we have constructed LDPC codes from α – resolvable BIB and Group divisible designs. The sub–matrices of incidence matrix of such block design are used as a parity – check matrix of the code which satisfy row – column constraint. Here the girth of the proposed code is at least six and the corresponding LDPC code (or Tanner graph) is free of 4– cycles.

For integers $n>2$ and $k>0$, an $(n\times n)/k$ semi-Latin square is an $n\times n$ array of $k$-subsets (called blocks) of an $nk$-set (of treatments), such that each treatment occurs once in each row and once in each column of the array. A semi-Latin square is uniform if every pair of blocks, not in the same row or column, intersect in the same positive number of treatments. We show that when a uniform $(n\times n)/k$ semi-Latin square exists, the Schur optimal $(n\times n)/k$ semi-Latin squares are precisely the uniform ones. We then compare uniform semi-Latin squares using the criterion of pairwise-variance (PV) aberration, introduced by J.P. Morgan for affine resolvable designs, and determine the uniform $(n\times n)/k$ semi-Latin squares with minimum PV aberration when there exist $n-1$ mutually orthogonal Latin squares (MOLS) of order $n$. These do not exist when $n=6$, and the smallest uniform semi-Latin squares in this case have size $(6\times 6)/10$. We present a complete classification of the uniform $(6\times 6)/10$ semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform $((n+1)\times (n+1))/((n-2)n)$ semi-Latin square when there exist $n-1$ MOLS of order $n$, and determine the PV aberration of such a uniform semi-Latin square. Finally, we describe how certain affine resolvable designs and balanced incomplete-block designs (BIBDs) can be constructed from uniform semi-Latin squares. From the uniform $(6\times 6)/10$ semi-Latin squares we classified, we obtain (up to block design isomorphism) exactly 16875 affine resolvable designs for 72 treatments in 36 blocks of size 12 and 8615 BIBDs for 36 treatments in 84 blocks of size 6. In particular, this shows that there are at least 16875 pairwise non-isomorphic orthogonal arrays $\mathrm{OA}(72,6,6,2)$., 21 pages; accepted manuscript

The issues concerning optimal estimation of unknown parameters in the model of chemical balance weighing designs with negative correlated errors are considered. The necessary and sufficient conditions determining the regular D‑optimal design and some new construction methods are presented. They are based on the incidence matrices of balanced incomplete block designs and balanced bipartite weighing designs. W artykule rozważa się problematykę dotyczącą istnienia regularnego D‑optymalnego chemicznego układu wagowego przy założeniu, że błędy pomiarów są ujemnie skorelowane i mają takie same wariancje. Przedstawiono warunki konieczne i dostateczne, wyznaczające układ regularnie D‑optymalny oraz podano nowe metody konstrukcji. Są one oparte na macierzach incydencji układów zrównoważonych o blokach niekompletnych oraz dwudzielnych układów bloków.

Youden Square Design (YSD) is an incomplete latin square design with at least one row/column which can’t run in an experiment. In this research we took 5x4 YSD (one column is not runned in an experiment). This design has a balance characteristic from a balanced incomplete block design where all treatments appears with the same number in each row. Missing data can occur in YSD. In this discussion, YSD with two missing data was used. Missing data is estimated by an iterative method then we arrange analysis of variance and LSD test. Analysis of variance with two missing data in YSD is calculated by adjusting the treatment sum of squares with it’s bias value and the total degrees of freedom and error degrees of freedom are substracted by two. LSD test is carried out if the treatment has a significant effect to the response. To clarify the discussion in YSD, example of application in the field of industry is given by observing the effect of the assembly method to the length of assembly time of X component. The assembly method has an effect to the length of assembly time of X component and if the missing data are and so the suggested assembly method is E method because it has the fastest average assembly time. Keywords: YSD, Missing Data, Analysis of Variance, LSD Test

The paper presents the results of research concerning on the estimation of individual weights of objects in spring balance weighing design. We consider D-optimal weighing design under assum...

Das and Gupta constructed optimal block designs for triallel crosses through Nested balanced block designs. In the present paper we are constructing three series of optimal partial triallel cross d...

This article deals with the optimality aspects of directional neighbor balanced block design under the assumption of randomness of block effects, both under additive and neighbor × direct effects n...

We study optimal block designs for comparing a set of test treatments with a control treatment. We provide the class of all E-optimal approximate block designs, which is characterized by simple lin...

International audience; In this paper we study the problem of finding neighbor optimal designs for a general correlation structure. We give universal optimality conditions for nearest-neighbor (NN) balanced block designs when observations on the same block are modeled by an autoregressive AR(m) process with arbitrary order m. The cases m=1,2 have been studied by Grondona and Cressie (Sankhyā Indian J Stat Ser A 55(2):267–284, 1993) for AR(2) and by Gill and Shukla (Biometrika 72(3):539–544, 1985a, Commun Stat Theory Methods 14(9):2181–2197, 1985b) and Kunert (Biometrika 74(4):717–724, 1987) for AR(1); we extend these results to the cases m≥3.

An accumulation of wireless sensor nodes is combined together to form the Wireless Sensor Networks. The sensor nodes are distributed haphazardly without any decided method into a natural setting, which is generally inhospitable and it is difficult to provide key-chains to each node for security as they are haphazardly distributed. In this paper, we use Nested Block Design (NBD) as Key Pre-distribution Scheme (KPS) and found out that NBD support large networks with fewer keys in each node than Symmetric Balanced Incomplete Block Designs (SBIBD) and Transversal Design (TD[Formula: see text]), provide higher resiliency than SBIBDs and better connectivity than TD[Formula: see text], tradeoff between local connectivity and resiliency ([Formula: see text]) is lower than SBIBD but more than TD[Formula: see text] and key-node ratio ([Formula: see text]) is same for [Formula: see text] but lower than SBIBD.

This paper derives a new randomization procedure by conditioning Efron’s (1971) biased coin design to a prespecified final balance. The new procedure remains a function of the original bias parameter which now controls the probability of intermediate balance rather than final balance. As the sample size increases, the design’s selection bias and intermediate balance are similar to those of the original biased coin, but unlike the original biased coin it always guarantees final balance. It is also shown that the permuted block design for equal allocation is a special case of the new procedure when used in blocks. The latter can substitute the permuted blocks with the added benefit of reducing the expected number of deterministic assignments. The new design is also noteworthy since it shows that a randomization procedure with new properties can be obtained by conditioning an existing one to a subset in its allocation space. New relationships among existing designs can be established in the process, further elucidating the protean nature of randomization.

Let G be a simple graph without isolated vertices. A graph H is said to be G-decomposable (or admit a G-decomposition), denoted by H → G, if the edges of H can be partitioned into edges of subgraphs isomorphic to G. A G-design is a G-decomposition of Kn, the complete graph with n vertices. If G is Kk, a G-design Kn → G is nothing more than a (n,k,1)-BIBD (balanced incomplete block design). G-designs can also be defined by using the concept of G-blocks, similar to the corresponding definition of a BIBD (see [5]).

Investigating (periodic) and design sequences with good correlation properties have numerous applications in communications. Research on designing sequence pairs with good correlation properties started in the early 1950's thanks to M.J. Golay. Ideally, one of our ultimate aims in this context is to design a set of sequences whose out-of-phase auto-correlation magnitudes and cross-correlation magnitudes are very small, preferably zero. The so-called Z-complementary pair (ZCP) is one of the suitable candidates. A pair of sequences is called a Z-complementary pair (ZCP) if it has zero aperiodic autocorrelation sums (AACSs) for time-shifts within a certain region, called zero correlation zone (ZCZ). ZCPs have been widely used in different communication systems and are closely related with almost difference families, which are useful in studying partially balanced incomplete block design. Despite remarkable progress in designing ZCPs, only a few constructions of quadriphase ZCPs (QZCPs) have been reported in the literature up to now. Aiming to reducing this gap, we explore in this article several methods to design such sequences. More specifically, we propose a recursive construction based on the concatenation of sequences aimed to design Type-II QZCPs. Also, based on Turyn's construction method, we present another new Type-II QZCPs. The proposed constructions lead to Z-optimal Type-II even-length QZCPs (E-QZCPs) and Type-II odd-length QZCPs (O-QZCPs) with large ZCZ widths. Finally, we derive upper bounds for the peak-to-mean envelope-power ratio (PMEPR) of the proposed ZCPs. It turns out that our constructions lead to ZCPs with low PMEPR. These characteristics allow our QZCPs to be seen as promising for practical uses in some modern communication systems.

In this article, we have discussed some aspects of block designs under systematic trend component. The necessary and sufficient condition for a block design to be trend free has been highlighted. A method of constructing linear trend free block designs in two plots per block have been discussed. The characterization properties of such designs have been investigated.

Recently, a construction of group divisible designs (GDDs) derived from the decoding of quadratic residue (QR) codes was given. In this paper, we extend the idea to obtain a new family of GDDs, which is also involved with a well-known balanced incomplete block design (BIBD).

In this paper we have proved that up to isomorphism there are exactly two orbit structures for a putative symmetric block design D with parameters (306,61,12), constructed by group G of order 61. Also the full automorphism groups for these orbit structures are given.

A block design is called feasible if all the factor effects in it are estimable. This paper studies the design structures of feasible block designs when the block sizes are unequal. Based on a necessary and sufficient condition, we find two structure features which respectively lead to unfeasible and feasible designs. According to these results, an effective way to find feasible designs is provided.

The present study aimed to assess the possible relationship between psychometric and electroencephalographic (EEG) parameters in humans. Normalized parameters of the integrated spatial (p S ) and integrated temporal (p T ) regularities of real resting EEG with the eyes closed were calculated. The algorithm of estimation of these parameters was based on the calculation of empirical multivariate temporal structure function of the first order. It was revealed that p S and p T parameters varied in a specific range, which was characterized by well-defined individual configuration reflecting the variability type (spatial, temporal, or mixed) of resting EEG. Obtained p S and p T parameters were compared to the speed of performed cognitive activity. The cognitive tasks included nonverbal subtests of the Wechsler Adult Intelligence Scale (WAIS) including block design according to a given template and construction of an object without any template, and motor probe. The successful performance on cognitive tasks was detected to be achieved with different combinations of p S and p T parameters in different subjects. Thus, the same intelligence resource can probably be organized with nonlinear principle in different subjects.

We propose a method for constructing a nested row–column design d 0 , involving a control treatment and v test treatments, starting from a Youden or Latin square T and an incomplete block design d. It is shown how the choice of T and d regulates the efficiency factors of d 0 . Special attention is given to the case when the v test treatments have a factorial structure, and for appropriately chosen d, an admissibility property of d 0 , among all comparable designs, is established. Illustrative examples are given.

In this paper, some aspects of design optimality on the basis of spring balance weighing designs are considered. The properties of D‑optimal and D‑efficiency designs are studied. The necessary and sufficient conditions determining the mentioned designs and some new construction methods are introduced. The methods of determining designs that have the required properties are based on a set of incidence matrices of balanced incomplete block designs and group divisible designs. W artykule zaprezentowane zostały problemy związane z optymalnością układu doświadczalnego z punktu widzenia sprężynowych układów wagowych. Przeanalizowano własności D‑optymalnych i wysoce D‑efektywnych układów. Podano warunki konieczne i dostateczne wyznaczające te układy. Ponadto zaprezentowano nową metodę konstrukcji tych układów, opartą na macierzach incydencji układów o grupach podzielnych oraz układów zrównoważonych o blokach niekompletnych.

Yantrams have been used to generate mixture designs in the interior of a simplex. In this note, we show a connection between Parshvanath yantram and a particular partially balanced incomplete block...