Symmetric Block Design Research Articles

Invariants of quadratic forms and applications in design theory

The study of regular incidence structures, such as projective planes and symmetric block designs, is a well established topic in discrete mathematics. The work of Bruck, Ryser and Chowla in the mid-twentieth century applied the Hasse-Minkowski local-global theory for quadratic forms to derive non-existence results for certain design parameters.Several combinatorialists have provided alternative proofs of this result, replacing conceptual arguments with algorithmic ones. In this paper, we show that the methods required are purely linear-algebraic and are no more difficult conceptually than the theory of the Jordan Canonical Form. Computationally, they are rather easier. We conclude with some classical and recent applications to design theory, including a novel application to the decomposition of incidence matrices of symmetric designs.

On the Symmetric Block Design with Parameters (210,77,28) Admitting a Frobenius Group of Order 57

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].

Symmetric Block Designs and Optimal Equidistant Codes

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.

Neighbor Discovery Optimization for Big Data Analysis in Low-Power, Low-Cost Communication Networks

Big data analysis generally consists of the gathering and processing of raw data and producing meaningful information from this data. These days, large collections of sensors, smart phones, and electronic devices are all connected in the network. One of the primary features of these devices is low-power consumption and low cost. Power consumption is one of the important research concerns in low-power, low-cost communication networks such as sensor networks. A primary feature of sensor networks is a distributed and autonomous system. Therefore, all network devices in this type of network maintain the network connectivity by themselves using limited energy resources. When they are deployed in the area of interest, the first step for neighbor discovery involves the identification of neighboring nodes for connection and communication. Most wireless sensors utilize a power-saving mechanism by powering on the system if it is off, and vice versa. The neighbor discovery process becomes a power-consuming task if two neighboring nodes do not know when their partner wakes up and sleeps. In this paper, we consider the optimization of the neighbor discovery to reduce the power consumption in wireless sensor networks and propose an energy-efficient neighbor discovery scheme by adapting symmetric block designs, combining block designs, and utilizing the concept of activating nodes based on the multiples of a specific number. The performance evaluation demonstrates that the proposed neighbor discovery algorithm outperforms other competitive approaches by analyzing the wasted awakening slots numerically.

Load-Optimal Local Fast Rerouting for Dense Networks

Reliable and highly available computer networks must implement resilient fast rerouting mechanisms: upon a link or node failure, an alternative route is determined quickly, without involving the network control plane. Designing such fast failover mechanisms capable of dealing with multiple concurrent failures, however, is challenging, as failover rules need to be installed proactively , i.e., ahead of time, without knowledge of the actual failures happening at runtime. Indeed, only little is known today about the design of resilient routing algorithms. This paper introduces a general framework to reason about and design local failover algorithms that minimize the resulting load after failover on dense networks, beyond destination-based routing. We show that due to the inherent locality of the failover decisions at runtime, the problem is fundamentally related to the field of distributed algorithms without coordination . We derive an intriguing lower bound on the inherent network load overhead any local fast failover scheme that will introduce in the worst case, even though globally seen, much more balanced traffic allocations exist. We then present different randomized and deterministic failover algorithms and analyze their overhead load. In particular, we build upon the theory of combinatorial designs and develop a novel deterministic failover mechanism based on symmetric block design theory , which tolerates a maximal number of link failures while ensuring low loads.

Self-dual codes from orbit matrices and quotient matrices of combinatorial designs

In this paper we give constructions of self-orthogonal and self-dual codes, with respect to certain scalar products, with the help of orbit matrices of block designs and quotient matrices of symmetric (group) divisible designs (SGDDs) with the dual property. First we describe constructions from block designs and their extended orbit matrices, where the orbit matrices are induced by the action of an automorphism group of the design. Further, we give some further constructions of self-dual codes from symmetric block designs and their orbit matrices. Moreover, in a similar way as for symmetric designs, we give constructions of self-dual codes from SGDDs with the dual property and their quotient matrices.

On the Symmetric Block Design with Parameters (306,61,12) Admitting a Group of Order 61

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.

Extended block designs for ideal system area networks

This paper considers a certain method of constructing 1-extended block designs; they generalize the concept of symmetrical block designs studied in combinatorial calculus. The definition of 1-extended block designs is provided and an algorithm of their construction is suggested. Finally, we analyze their application for building distributed full switches as "ideal" system area networks in multiprocessor computer systems.

작은 입력신호를 위한 Two-Dimensional Symmetric Balance Incomplete Block Design Code

작은 입력신호의 spectral-amplitude-code(SAC) optical code-division multiple-access (OCDMA) 시스템에는 nonideal symmetric balance incomplete block design(BIBD) code의 사용이 효율적이나 충분한 사용자를 수용하지 못하는 단점이 있다. 이를 극복하기 위하여 본 논문에서는 ideal BIBD code를 공간 코드로 사용하고 nonideal code를 파장 코드로 사용하는 two-dimensional(2-D) BIBD code를 제안한다. 사용자 수에 따른 bit error-rate(BER) 분석을 통하여 제안하는 2-D BIBD code가 1-D BIBD code에 비하여 최대사용자 수를 현저하게 증가시킬 수 있음을 알 수 있다.

Symmetric Balance Incomplete Block Design Code의 Spectral Efficiency

본 논문은 symmetric balance incomplete block design(BIBD) code의 BER=<SUP>-9</SUP>을 만족하는 spectral efficiency를 구하였다. 이 계산 결과 effective power가 큰 경우 (P<SUB>sr</SUB>=-10 dBm)는 m=2로 고정시키고 q값을 변화시키는 ideal BIBD code 구성이 효율적이었다. 이와 반대로 effective power가 작은 경우 (P<SUB>sr</SUB>=-25 dBm)는 ideal BIBD code 구성 보다는 q>2인 값을 취하고 m값을 변화시키는 설계가 더 효율적임을 알 수 있었다.

Is Treatment Adjustment in SBIBD Necessary

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.

All λ‐Designs with Small λ are Type‐1

AbstractA λ‐design is a family of subsets of such that for all and not all are of the same size. Ryser's and Woodall's λ‐design conjecture states that each λ‐design can be obtained from a symmetric block design by a certain complementation procedure. Our main result is that the conjecture is true when λ &lt; 63. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 408–431, 2012

Query-enabled sensor networks using the cyclic symmetric wakeup design

The problem of conserving energy under multiple constraints of query waiting delays and sensing coverage preservation in query-based wireless sensor networks is addressed. We propose a network architecture based on the cyclic symmetric block design class of wakeup schemes. Using a target tracking application as an example, specific requirements of the application translates to design parameters in our proposed solution. Our proposal demonstrates good delay performances, high target tracking accuracies and low target identification errors, without the need for costly double-radio channels while increasing network lifetimes by a factor of four to eight over traditional methods.

Combinatorial methods for constructing bipartite uniform minimal quasicomplete graphs (symmetrical block designs)

Consideration was given to a specific family of bipartite graphs consisting of two disjoint subsets X and Y of vertices and characterized by that each vertex in X (Y) is connected to each of the remaining vertices in X (Y) by a unique path of length two passing through some vertex in Y (X). The prefix "quasi" reflects the fact that complete connection of the vertices is realized by paths of length two rather than by edges. The problem of constructing uniform minimal graphs with identical cardinalities of the subsets X and Y which is of practical interest for complex communication networks was discussed. It belongs to the class of combinatorial problems of construction of the so-called symmetrical block designs.

Block designs which are divisible in multiple ways

AbstractConstructions are given for the production of symmetric block designs whose point set is divisible (partitionable) in several ways. Some properties of the dual design are also considered. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 415–422, 2006

A classification of all symmetric block designs of order nine with an automorphism of order six

AbstractWe complete the classification of all symmetric designs of order nine admitting an automorphism of order six. As a matter of fact, the classification for the parameters (35,17,8), (56,11,2), and (91,10,1) had already been done, and in this paper we present the results for the parameters (36,15,6), (40,13,4), and (45,12,3). We also provide information about the order and the structure of the full automorphism groups of the constructed designs. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 301–312, 2006

Ordinary graphs and subplane partitions

We introduce a generalization of symmetric ( v, k, λ) block designs, and show how these could potentially be used to construct projective planes of nonprime-power order.

Centering small generalized polygons–projective pottery at work

The generalized triangle, quadrangle, and hexagons of order 2 are small point-line geometries that play a role in the theory of generalized polygons and buildings that is comparable to that of the Fano plane in the theory of projective planes. Virtually everybody working in discrete mathematics is familiar with the generalized triangle of order 2 aka the Fano plane, the smallest projective plane, PG(2,2), or the unique symmetric block design with parameters 2−(7,3,1), and its elementary representation as the geometrical configuration of an equilateral triangle together with its three medians and inscribed circle. The main purpose of this paper is to derive an elementary representation of the generalized hexagons of order 2 which extends naturally to a special representation of the projective space PG(5,2). Along the way, we also derive a similar representation of the generalized quadrangle of order 2 which turns out to be equivalent to the well-known representation of the quadrangle in terms of synthems and duads and which extends naturally to a representation of PG(3,2).

Construction of symmetric block designs using normal subgroups of automorphism groups

Construction of symmetric block designs using normal subgroups of automorphism groups

An introduction to cocyclic generalised Hadamard matrices

Many codes and sequences designed for robust or secure communications are built from Hadamard matrices or from related difference sets or symmetric block designs. If an alphabet larger than {0,1} is required, the natural extension is to generalised Hadamard matrices, with entries in a group. The code and sequence construction techniques for Hadamard matrices are applicable to the general case. A cocyclic generalised Hadamard matrix with entries in an abelian group is equivalent to a semiregular central relative difference set and to a divisible design with a regular group of automorphisms, class regular with respect to the forbidden central subgroup. In this introduction we outline the necessary background on cocycles and their properties, give some familiar examples of this unfamiliar concept and demonstrate the equivalence of the above-mentioned objects. We present recent results on the theory of cocyclic generalised Hadamard matrices and their applications in one area: error-correcting codes.