Balanced Ternary Designs Research Articles

Modified Slope Rotatability for Balanced Ternary Designs of Second Order Response Surfaces

Objective: To evaluate the best optimal solutions of modified Second Order Slope Rotatable Designs (SOSRD) using Balanced Ternary Designs (BTD) and to compare with different methods of Modified SOSRD for 3≤v≤8. Methods: To estimate the parameters of the response surface design of second order the principle of least squares was applied for design of experiments, we also evaluated modified slope rotatable value and variance of estimated responses. Findings: The outcome of this research reveals the influence of various design parameters on key aspects such as the variance of the estimated response, slope rotatable value, axial points, central points and also total number of design points of modified SOSRD framework. Novelty: The variance of the estimated responses in the new method of Modified SOSRD using BTD provides the best optimal solutions with minimum variances while compared to different methods of Modified SOSRD. Keywords: Response ­ Surface Experimental Designs, Second Order Rotatable Designs, Second ­ Order Slope ­ Rotatability, Second ­ Order Modified Slope ­ Rotatability, Balanced Ternary Designs

Measure of Rotatability for a Class of Balanced Ternary Design

In the present study we mainly focused on a measure of rotatability for a class of Balanced ternary designs. It is very much useful to assess the degree of a rotatability for a given response surface design.

New class of second order rotatable designs using balanced ternary designs

In this paper, following works of Tyagi and Rizwi (1979), Kanna et al. (2018), a new class of second order rotatable designs (SORD) using balanced ternary designs are suggested. Here we develop a method of balanced ternary designs by combining the methods of Tyagi and Rizwi. Here we observed that the number of design point is lesser than the other designs. Table 3 provides the complete idea regarding the smaller number of designs points when we compare with the balanced incomplete block designs, Tyagi and Rizwi design points.

Combined Intra- and Inter- Block Analysis of Balanced Ternary Designs

In the present paper an attempt has been made to study the inter—and intra-block estimation of treatment effects contrasts with random block effects in the context of balanced ternary designs. The analysis is illustrated through examples.

Some Constructions of Balanced Ternary Residual Treatment Effects Designs

In balanced ternary residual treatment effects designs, a treatment occurs in a given sequence at most twice but the same treatment does not appear in two consecutive periods. This paper gives some methods of construction of new families of balanced ternary residual treatment effects designs.

Construction of some classes of balanced cross-over designs from terraces

Anderson and Preece (2003a, 2003b) have recently introduced power-sequence terraces. In this paper we have used these terraces for the construction of some new families of balanced cross-over designs of first and second order which are variance-balanced. We have also used them for the construction of some new families of balanced ternary cross-over designs.

Enumeration of balanced ternary designs

A ( V, B; ρ 1, ρ 2, R; K,Λ) balanced ternary design is a pair ( V , B ) , where V is a V-set of points and B is a collection of B K-multisubsets of V called blocks, such that each point appears R times in the blocks and no block contains a point with multiplicity greater than two. Each point must appear in ρ 1 blocks with multiplicity one and in ρ 2 blocks with multiplicity two. Additionally, every pair of distinct points must appear exactly Λ times in the blocks of the design. A backtrack search algorithm with isomorph rejection is described and employed to enumerate the balanced ternary designs with V⩽10, B⩽30, and R⩽15 for all but 12 of the 155 possible design classes with these parameters.

Balanced Incomplete Block Designs with Block Size 9 and λe2,4,8

The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: \begin{array}{rcl} \lambda (v - 1) &\equiv& 0\ \mod (k - 1) \\[2pt] \lambda v(v - 1 ) &\equiv& 0\ \mod k(k - 1) \end{array} In this paper we study ke9 with λe2,4 or 8. For λe8, we show these conditions on v are sufficient, and for λe2, 4 respectively there are 8 and 3 possible exceptions the largest of which are ve1845 and 783. We also give some examples of group divisible designs derived from balanced ternary designs.

Construction of nested balanced block designs, rectangular designs andq-ary codes

Construction of nested balanced incomplete block (BIB) designs, nested balanced ternary designs and rectangular designs from given nested BIB designs and resolvable BIB designs are described. New constructions ofq-ary codes from nested BIB designs and balanced bipartite weighing designs are also given.

Construction of Balanced Ternary Designs with Nested Rows and Columns Using BIB Designs

Balanced ternary designs with nested rows and columns have been constructed using balanced incomplete block designs.

Symmetric balanced ternary designs with ϱ1 = 1 or 2

We prove the following two non-existence theorems for symmetric balanced ternary designs. If ϱ1 = 1 and Λ ≡ 0 (mod 4) then eitherV = Λ + 1 or 4ϱ2 − Λ + 1 is a square and √(4ϱ2 − Λ + 1) divides Λ2 − 1. If ϱ1 = 2 thenV = ((m + 1)/2)2 + 2,K = (m2 + 7)/4 and Λ = ((m − 1)/2)2 + 1 wherem ≡ 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.

Symmetric balanced ternary designs with ϱ1=1 or 2

Symmetric balanced ternary designs with ϱ1=1 or 2

Balanced ternary designs with block size four

Abstract A balanced ternary design with block size four is a collection of multi-sets of size four, chosen from a v -set in such a way that each element of the v -set occurs 0, 1 or 2 times in each of the multi-sets (which are called blocks). Also each distinct pair of elements occurs λ times (λ is the index) and each element is repeated (that is, occurs twice) in ϱ 2 blocks. (Here, a block such as { a,a,b,c }, written aabc for short, is said to contain six pairs: aa and bc once each, and ab and ac twice each.) In this paper we give necessary and sufficient conditions for the existence of a balanced ternary design with block size 4, any indexλ, and ϱ 2 ⩽6.

Constructions of balanced ternary designs based on generalized Bhaskar Rao designs

New series of balanced ternary designs and partially balanced ternary designs are obtained. Some of the designs in the series are non-isomorphic solutions for design parameters which were previously known or whose solution was obtained by trial and error, rather than by a systematic method.

The number of repeated blocks in balanced ternary designs with block size three II

Let D denote any balanced ternary design on v elements with block size three, index two, and ϱ 2 = 2 (so each element occurs repeated in precisely two blocks). D is thus also a multi-set design MB 2(υ, 3, 2), in the terminology of Assaf, Hartman and Mendelsohn. This paper shows that such a design D exists which contains exactly k pairs of repeated blocks if and only if υ ≡ 0 or 2 modulo 3, υ ⩾ 5 and 0 ⩽ k ⩽ t υ = 1 16 υ(υ − 5), k ≠ t υ − 1 .

Constructions of balanced ternary designs

AbstractA construction for balanced ternary designs is given. Based on the designs so obtained, a construction of partially balanced ternary designs is given, which gives balanced ternary designs and series of symmetric balanced ternary designs in special cases.

Some Constructions of Partially Balanced Ternary Design*

This paper gives some constructions of Partially Balanced Ternary (PBT) designs, starting from a given linked block design.

Topics in balanced ternary designs

In this thesis, we investigate the properties of balanced ternary designs.We begin with a survey of the literature on balanced n-ary designs, of which balanced ternary designs are a special case. This provides a setting for our research as well as the necessary background information.Our main objective is to establish new existence and enumeration results. We develop methods for constructing certain balanced ternary designs. In some cases we construct entire classes of balanced ternary designs with block size four; hence obtaining necessary and sufficient conditions for their existence. We also enumerate the number of non-isomorphic balanced ternary designs with certain sets of parameters.A number of the papers surveyed state conditions which place constraints on the parameter sets of balanced ternary designs. We find additional conditions which must be satisfied by the parameters of some balanced ternary designs. As a result^we prove the non-existence of two balanced ternary designs, the existence of which was left open by Billington and Robinson.In the survey we document papers which give methods for constructing balanced n-ary designs from balanced incomplete block designs, affine a-resolvable balanced incomplete block designs, partially balanced incomplete block designs, group divisible designs, finite geometries, difference sets and cyclotomic classes, and Latin squares.We give new results which construct balanced ternary designs from the incidence matrices of balanced incomplete block designs and from the blocks of a 1-design. We show that balanced ternary designs, with block size a prime or prime power, can be embedded into the set of blocks of another balanced ternary design, thereby establishing the existence of this second design. We give a recursive construction which combines the blocks of a group divisible design with the blocks of s frames and the blocks of an existing balanced ternary design to form the set of blocks of a 'new' balanced ternary design. This construction is analogous to that used by Hanani to construct balanced incomplete block designs.We survey papers which give necessary and sufficient conditions for the existence of balanced ternary designs with block size three and some balanced ternary designs with block size four. We extend this work and go on to give a detailed study of balanced ternary designs with block size four. We construct a number of individual designs^ as well as an infinite family of cyclic balanced ternary designs, with block size four. These designs are then used in conjunction with the above recursive construction to construct entire families of balanced ternary designs, with block size four, index 2 and r2 = 1,...,6, where r2 denotes the number of blocks in which an element occurs twice. In the case of r = 2;, we complete recent work by Assaf, Hartman and Mendelsohn. We are then in a position to state necessary and sufficient conditions for the existence of such designs. Balanced ternary designs with block size four, any index and any r2 are discussed.Finally, we consider the enumeration of some balanced ternary designs. Certain sets of parameters are considered, and the exact number of non-isomorphic balanced ternary designs, with these parameters, are given. Quasi-T-multiples of balanced ternary designs are discussed, and their reducibility or irreducibility is considered.

More balanced ternary designs with block size four

This paper gives necessary and sufficient conditions for the existence of balanced ternary designs with block size four and Λ = 2 in the cases ϱ 2 = 3, 4, 5 and 6; the case ϱ 2 = 1 and 2 have appeared earlier.

A family of balanced ternary designs with block size four

This paper shows the existence of an infinite family of cyclic balanced ternary designs where the block size is 4, the index 2 and each block contains precisely one repeated element.