Resolvable Balanced Incomplete Block Designs Research Articles

On magic total labelings in combinatorial designs

Let H be a graph and ẞ be an H-decomposition of G. The H-magic total labeling of a graph G is an assignment of integers set [1, |V(G)∪E(G)|] to V(G)∪E(G) such that: any vertex and any edge of G receive distinct integers, and the sum of all vertices receive integers and all edges receive integers on B is a constant for any B ∈ ẞ. In this paper we study the H-super magic problems of balanced incomplete block designs and resolvable balanced incomplete block designs.

Optimal key pre‐distribution schemes from affine resolvable and partially affine resolvable designs

AbstractVarious key pre‐distribution schemes (KPSs) for distributed sensor networks (DSNs) were obtained from different combinatorial designs. A block design may be mapped to a KPS where is the number of nodes or network size and is the number of keys (points) per node. A KPS is corresponding to a symmetric balanced incomplete block design has full connectivity where all nodes are connected to each other. But it is desirable that all nodes do not share common keys as the compromise of any node will destroy the whole network (see Lee & Stinson, 2005; Martin, 2009). To meet the situation, Lee and Stinson (2005) introduced μ—common intersection design which has disjoint blocks and these disjoint blocks can establish a secure communication through certain common intersecting blocks. Here we have proposed optimal KPSs from affine resolvable and partially affine resolvable balanced incomplete block designs, group divisible designs and L2—type designs. It is shown that these designs are important classes of μ—common intersection designs. The KPSs obtained here are optimal in the sense of local connectivity. Mathematics Subject Classification (2010): 94C30; 68R05; 05B05; 62K10.

THE INTERSECTION PROBLEM FOR RESOLVABLE BALANCED INCOMPLETE BLOCK DESIGNS WITH SMALL ORDERS

The intersection problem is one of the basic problems in combinatorial design theorys. In this paper, two cases of resolvable balanced incomplete block designs of size four are investigated. As the basic step of the study on the intersection problem for resolvable balanced incomplete block designs, we have determined some values of the intersection numbers. It will play an important role on the further recursive constructions.

A method of constructing optimal sliced uniform designs under discrete discrepancy

Sliced uniform designs are useful for generating experimental data with batch structures. In this paper, we employ the discrete discrepancy as the measure of uniformity to construct sliced uniform designs. The construction method is based on an important class of combinatorial configurations, namely, resolvable balanced incomplete block designs; hence, the construction method does not require any computer search. Through resolvable balanced incomplete block designs, under the discrete discrepancy, several infinite classes of new sliced uniform designs are obtained. The obtained design is optimal in the sense that not only the design as a whole but also each of its slices attain the lower bound of the discrete discrepancy. The design parameters are explicitly given, from which experimenters may choose an optimal sliced uniform design with the most suitable run size for their experiment.

A SIMPLE GENERALIZED CONSTRUCTION OF RESOLVABLE BALANCED INCOMPLETE BLOCK DESIGNS WITH PRIME BLOCK SIZES

This paper presents a Simple Generalized Construction of Resolvable Balanced Incomplete Block Designs whose parameter combination is of the form $v=k^2, ~r=k+1, ~\lambda=k^0=1$, where $k$ is prime. The design construction was achieved by using the cyclic subgroup of the symmetric group $S_k$ whose generator is one of the permutations of the $2$-permutation generating set of the Dihedral group $D_k$ and $2$-permutation generating set of the presentation of $S_k$. The method is efficient, sufficient and also mitigate against the tediousness encountered in other methods of construction when $v$ is large.

On traceable results of linear error correcting codes and resolvable BIBDS

In this paper we relate how Equidistant Constant Weight Codes and Different Combinatorial Structures like Resolvable Balanced Incomplete Block Designs (RBIBD) , Nested Balanced Incomplete Block Designs (NBIBD) and Linear Codes are related with each other and then show how these Combinatorial Structures can be used as 2-Traceable (TA) Code.

Corrigendum: Super‐simple resolvable balanced incomplete block designs with block size 4 and index 2

AbstractIn this note, we present a corrected proof of Lemma 5.8 which was wrong due to the incorrectness of Construction 2.3, from our paper.

Frame difference families and resolvable balanced incomplete block designs

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational $$(pq+1,p+1,1)$$ -RBIBDs over $${\mathbb {F}}_{p}^+ \times {\mathbb {F}}_{q}^+$$ are derived, and the existence of $$(125q+1,6,1)$$ -RBIBDs is discussed. We construct (v, 8, 1)-RBIBDs for $$v\in \{624,$$ $$1576,2976,5720,5776,10200,14176,24480\}$$ , whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences.

New Diallel Cross Designs Through Resolvable Balanced Incomplete Block Designs for Field Experiments

New Diallel Cross Designs Through Resolvable Balanced Incomplete Block Designs for Field Experiments

Zig-Zag matrix for resolvable nested balanced incomplete block designs

This paper presents a method of construction of designs which utilizes special matrix structures referred to as Zig-zag. The Zig-zag matrix structures give rise to initial blocks for resolvable nested balanced incomplete block designs (RNBIBDs). The construction of the designs for \((v)\) treatment number being a perfect square with specific interest on \((k)\) block size as a prime number are presented. The designs constructed are of high efficiency with minimum blocks. A generalized information matrix is also obtained.

On generalized Howell designs with block size three

In this paper, we examine a class of doubly resolvable combinatorial objects. Let $$t, k, \lambda , s$$t,k,ź,s and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t-$$\mathrm {GHD}_{k}(s,v;\lambda )$$GHDk(s,vźź), is an $$s\times s$$s×s array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of X); (ii) no t-subset of elements from X appears in more than $$\lambda $$ź cells. Particular instances of the parameters correspond to Howell designs, doubly resolvable balanced incomplete block designs (including Kirkman squares), doubly resolvable nearly Kirkman triple systems, and simple orthogonal multi-arrays (which themselves generalize mutually orthogonal Latin squares). Generalized Howell designs also have connections with permutation arrays and multiply constant-weight codes. In this paper, we concentrate on the case that $$t=2$$t=2, $$k=3$$k=3 and $$\lambda =1$$ź=1, and write $$\mathrm {GHD}(s,v)$$GHD(s,v). In this case, the number of empty cells in each row and column falls between 0 and $$(s-1)/3$$(s-1)/3. Previous work has considered the existence of GHDs on either end of the spectrum, with at most 1 or at least $$(s-2)/3$$(s-2)/3 empty cells in each row or column. In the case of one empty cell, we correct some results of Wang and Du, and show that there exists a $$\mathrm {GHD}(n+1,3n)$$GHD(n+1,3n) if and only if $$n \ge 6$$nź6, except possibly for $$n=6$$n=6. In the case of two empty cells, we show that there exists a $$\mathrm {GHD}(n+2,3n)$$GHD(n+2,3n) if and only if $$n \ge 6$$nź6. Noting that the proportion of cells in a given row or column of a $$\mathrm {GHD}(s,v)$$GHD(s,v) which are empty falls in the interval [0, 1 / 3), we prove that for any $$\pi \in [0,5/18]$$źź[0,5/18], there is a $$\mathrm {GHD}(s,v)$$GHD(s,v) whose proportion of empty cells in a row or column is arbitrarily close to $$\pi $$ź.

Optimal almost equisymbol codes based on difference matrices

A method for constructing optimal almost equisymbol codes based on difference matrices is proposed. Some of these codes generate previously unknown optimal equidistant codes satisfying the Plotkin bound. As a consequence, they generate new resolvable balanced incomplete block designs.

On the Enumeration of ‐Optimal and Minimax‐Optimal k‐Circulant Supersaturated Designs

AbstractIn recent years, several methods have been proposed for constructing ‐optimal and minimax‐optimal supersaturated designs (SSDs). However, until now the enumeration problem of such designs has not been yet considered. In this paper, ‐optimal and minimax‐optimal k‐circulant SSDs with 6, 10, 14, 18, 22, and 26 runs, factors and are enumerated in a computer search. We have also enumerated all ‐optimal and minimax‐optimal k‐circulant SSDs with (mod 4) and . The computer search utilizes the fact that theses designs are equivalent to certain 1‐rotational resolvable balanced incomplete block designs. Combinatorial properties of these resolvable designs are used to restrict the search space.

Super-Simple Resolvable Balanced Incomplete Block Designs with Block Size 4 and Index 4

The necessary conditions for the existence of a super-simple resolvable balanced incomplete block design on v points with block size k = 4 and index ? = 4, are that v ? 16 and v ? 4 (mod 12). These conditions are shown to be sufficient.

From a 1-Rotational RBIBD to a Partitioned Difference Family

Generalizing the case of $\lambda=1$ given by Buratti and Zuanni [Bull Belg. Math. Soc. (1998)], we characterize the $1$-rotational difference families generating a 1-rotational $(v,k,\lambda)$-RBIBD, that is a $(v,k,\lambda)$ resolvable balanced incomplete block design admitting an automorphism group $G$ acting sharply transitively on all but one point $\infty$ and leaving invariant a resolution $\cal R$ of it. When $G$ is transitive on $\cal R$ we prove that removing $\infty$ from a parallel class of $\cal R$ one gets a partitioned difference family, a concept recently introduced by Ding and Yin [IEEE Trans. Inform. Theory, 2005] and used to construct optimal constant composition codes. In this way, by exploiting old and new results about the existence of 1-rotational RBIBDs we are able to derive a great bulk of previously unnoticed partitioned difference families. Among our RBIBDs we construct, in particular, a $(45,5,2)$-RBIBD whose existence was previously in doubt.

Classification of resolvable balanced incomplete block designs — the unitals on 28 points

Abstract Approaches for classifying resolvable balanced incomplete block designs (RBIBDs) are surveyed. The main approaches can roughly be divided into two types: those building up a design parallel class by parallel class and those proceeding point by point. With an algorithm of the latter type — and by refining ideas dating back to 1917 and the doctoral thesis by Pieter Mulder — it is shown that the list of seven known resolutions of 2-(28, 4, 1) designs is complete; these objects are also known as the resolutions of unitals on 28 points.

Resolvable balanced incomplete block designs with subdesigns of block size 4

In this paper, we look at resolvable balanced incomplete block designs on v points having blocks of size 4, briefly ( v , 4 , 1 ) RBIBDs. The problem we investigate is the existence of ( v , 4 , 1 ) RBIBDs containing a ( w , 4 , 1 ) RBIBD as a subdesign. We also require that each parallel class of the subdesign should be in a single parallel class of the containing design. Removing the subdesign gives an incomplete RBIBD, i.e., an IRB ( v , w ) . The necessary conditions for the existence of an IRB ( v , w ) are that v ⩾ 4 w and v ≡ w ≡ 4 ( mod 12 ) . We show these conditions are sufficient with a finite number (179) of exceptions, and in particular whenever w ≡ 16 ( mod 60 ) and whenever w ⩾ 1852 . We also give some results on pairwise balanced designs on v points containing (at least one) block of size w , i.e., a ( v , { K , w * } , 1 ) -PBD. If the list of permitted block sizes, K 5 , contains all integers of size 5 or more, and v , w ∈ K 5 , then a necessary condition on this PBD is v ⩾ 4 w + 1 . We show this condition is not sufficient for any w ⩾ 5 and give the complete spectrum (in v ) for 5 ⩽ w ⩽ 8 , as well as showing the condition v ⩾ 5 w is sufficient with some definite exceptions for w = 5 and 6 , and some possible exceptions when w = 15 , namely 77 ⩽ v ⩽ 79 . The existence of this PBD implies the existence of an IRB ( 12 v + 4 , 12 w + 4 ) . If the list of permitted block sizes, K 1 ( 4 ) , contains all integers ≡ 1 ( mod 4 ) , and v , w ∈ K 1 ( 4 ) , then a necessary condition on this PBD is v ⩾ 4 w + 1 . We show this condition is sufficient with a finite number of possible exceptions, and in particular is sufficient when w ⩾ 1037 . The existence of this PBD implies the existence of an IRB ( 3 v + 1 , 3 w + 1 ) .

Super‐simple resolvable balanced incomplete block designs with block size 4 and index 2

AbstractThe necessary conditions for the existence of a super‐simple resolvable balanced incomplete block design on v points with block size k = 4 and index λ = 2, are that v ≥ 16 and $v \equiv 4\; (\bmod\; {12})$. These conditions are shown to be sufficient. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 341–356, 2007

On balanced incomplete-block designs with repeated blocks

Balanced incomplete-block designs (BIBDs) with repeated blocks are studied and constructed. We continue work initiated by van Lint and Ryser in 1972 and pursued by van Lint in 1973. We concentrate on constructing ( v , b , r , k , λ ) -BIBDs with repeated blocks, especially those with gcd ( b , r , λ ) = 1 and r ≤ 20 . We obtain new bounds for the multiplicity of a block in terms of the parameters of a BIBD, and improvements to these bounds for a resolvable BIBD. This allows us to answer a question of van Lint about the sufficiency of certain conditions for the existence of a BIBD with repeated blocks.

General frame constructions for Z-cyclic triplewhist tournaments

Frames are useful in dealing with resolvable designs such as resolvable balanced incomplete block designs and triplewhist tournaments. Z-cyclic triplewhist tournament frames are also useful in the constructions of Z-cyclic triplewhist tournaments. In this paper, the concept of an ( h 1 , h 2 , … , h n ; u ) -regular Z-cyclic triplewhist tournament frame is defined, and used to establish several quite general recursive constructions for Z-cyclic triplewhist tournaments. As corollaries, we are able to unify many known constructions for Z-cyclic triplewhist tournaments. As an application, some new Z-cyclic triplewhist tournament frames and Z-cyclic triplewhist tournaments are obtained. The known existence results of such designs are then extended.