Efficiency of Sum Constructed Automorphic Symmetric Balanced Incomplete Block Designs

An arrangement of $v$ varieties or treatments in $b$ blocks of size $k, (k < v),$ is known as a balanced incomplete block design if every variety occurs in $r$ blocks and any two varieties occur together in $\\lambda$ blocks. These parameters obviously satisfy the equations \\begin{equation*}\\tag{1} bk = vr\\end{equation*}\\begin{equation*}\\tag{2} \\lambda(v - 1) = r(k - 1)\\end{equation*} Fisher [1] has also proved that the inequality \\begin{equation*}\\tag{3} b \\geq v, \\quad r \\geq k\\end{equation*} must hold. If $v, b, r, k$ and $\\lambda$ are positive integers satisfying (1), (2) and (3), then a balanced incomplete block design with these parameters possibly exists, but the actual existence of a combinatorial solution is not ensured. These conditions are thus necessary but not sufficient for the existence of a design. Fisher and Yates in their tables [2] have listed all designs with $r \\leq 10$ and given combinatorial solutions, where known. A balanced incomplete block design in which $b = v$, and hence $r = k$ is called a symmetrical balanced incomplete block design. The impossibility of the symmetrical designs with parameters $v = b = 22, r = k = 7, \\lambda = 2$ and $v = b = 29, r = k = 8, \\lambda = 2$ was first demonstrated by Hussain [3], [4] essentially by the method of enumeration. The object of the present note is to give an alternative simple proof of the impossibility of these designs and to show that the only unknown remaining symmetrical design in Fisher and Yates' tables, viz. $v = b = 46, r = k = 10, \\lambda = 2,$ is definitely impossible. Symmetrical designs with $\\lambda \\leq 5, r, k \\leq 20$, which are impossible combinatorially, are also listed.

The paper discusses the symmetric balanced incomplete block designs (SBIBD’s) and further verifies whether treatment adjustment in SBIBD is necessary using analysis of variance (ANOVA) method. Fortunately, the data collected for the analysis were in the same group of the balanced incomplete-block designs (BIBD’s), that is, symmetric balance incomplete block design (SBIBD).The ANOVA method is applied differently on the two sets of data arising from the symmetric balanced incomplete block designs (SBIBDs). In this work, the basic interest is to verify empirically whether treatment adjustment is necessary in SBIBDs with data from National Root Crop Research Institute (N.R.C.R.I), Umudike, AbiaState. To achieve this, we considered treatment (not adjusted), treatment (adjusted), and block (adjusted) in the ANOVA method. The study discovered that if ANOVA method is applied on the symmetric balanced incomplete block design (SBIBD) when a treatment is not significant, it turns out to be significant when the same treatment is adjusted. It obvious from the experiments that were carried out that the adjustment of the treatment in SBIBD’s is very necessary when the ANOVA method is applied. Apart from the improvement in the level of significance of the result, the error is also greatly reduced when treatment is adjusted.

Motivated by the field experimental designs in agriculture, the theory of block designs has been applied to several areas such as statistics, combinatorics, communication networks, distributed systems, cryptography, etc. An explicit formula and its fast computational algorithm for a class of symmetric balanced incomplete block designs are presented. Based on the formula and the careful investigation of the modulus multiplication table, the algorithm is developed. The computational costs of the algorithm is superior to those of the conventional ones.

The design and analysis of an efficient load balancing algorithm employing the symmetric balanced incomplete block design

Cloud computing is one of the recent emerging technologies. Heavy data sharing among multiple users is an open issue. Data sharing in cloud computing enables multiple participants to freely share the group data, which improves the efficiency of work in cooperative environments and has widespread potential applications. However, how to ensure the security of data sharing within a group and how to efficiently share the outsourced data in a group manner are formidable challenges. In this paper we focused on security issues with the help of key agreement protocols to perform efficient data sharing in cloud environment. We proposed a novel block design based key agreement method in support of symmetric balanced incomplete block design (SBIBD). The main objective of this method is to supports multiple participants, which can flexibly extend the number of participants in a cloud environment according to the structure of the block design. Based on the proposed group data sharing model, we present general formulas for generating the common conference key K for multiple participants. In addition, the fault tolerance property of our protocol enables the group data sharing in cloud computing to withstand different key attacks, which is similar to Yi’s protocol.

: It is shown that among binary and connected block designs with and without equal block sizes and equal replications only a symmetric balanced incomplete block design produces a completely symmetric information matrix for the treatment effects whenever the number of blocks is equal to the number of treatments and the number of experimental units is a multiple of the number of treatments. This result contains a known result as a special case.

Some new construction methods of the optimum chemical balance weighing designs and pairwise efficiency and variance balanced designs are proposed, which are based on the incidence matrices of the known symmetric balanced incomplete block designs. Also the conditions under which the constructed chemical balance weighing designs become A-optimal are also been given.

BCH codes and Designs

Symmetric balanced incomplete block designs (SBIBD) and Hadamard matrices

All the symmetric balanced incomplete block (SBIB) designs have been characterized and a new generalized expression on parameters of SBIB designs has been obtained. The parameter b has been formulated in a different way which is denoted by b i , i = 1, 2, 3, associating with the types of the SBIB design D i . The parameters of all the designs obtained through this representation have been tabulated while corresponding them with the suitable formulae for the number of blocks b i and the expression S i for the convenience of practical users for constructional methods of certain designs, which is the main theme of this paper.

A Machine learning approach to predict re-key in wireless sensor network using symmetric balanced incomplete block designs

Abstract This paper illustrates, through a special case, an effect of non-independent numerators on F-test's in an analysis of variance. Symmetrical balanced incomplete block designs are considered with "blocks" a random effect and no interaction. The joint density of the adjusted sums of squares is determined under the null hypothesis of no treatment effect. Given the result of testing for the block effect, the conditional probability of a Type I error when testing the null hypothesis concerning the treatment effect is evaluated for a number of designs.

An identity-based conference key distribution protocol based on SBIBD was proposed by Chung et al. in 2002 [1]. However, this protocol suffers several problems. Consequently, participants cannot correctly authenticate one another and cannot calculate the same conference key. This study investigates these problems and proposes an improved protocol also based on SBIBD.

Tables of Parameters of BIBDs with r⩽41 including Existence, Enumeration, and Resolvability Results

Kim and Ko proposed second type second order slope rotatable designs (SOSRD) utilizing central composite designs (CCD) wherein the two digits   1 2 a , a denotes the position of star points. In this study, we propose SOSRD of second type utilizing balanced incomplete block designs (BIBD) with unequal block sizes. In specific cases, the recommended procedure results in fewer design points than SOSRD of second type acquired through pairwise balanced designs (PBD), symmetrical unequal block arrangements (SUBA) with two unequal block sizes and balanced incomplete block designs (BIBD).