Group Divisible Designs Research Articles

Incomplete group divisible designs with block size four and general index

In this paper, we investigate the existence of incomplete group divisible designs (IGDDs) with block size four, group-type (g, h)u and general index λ. The necessary conditions for the existence of such a design are that u ≥ 4, g ≥ 3h, λg(u − 1) ≡ 0 (mod 3), λ(g − h)(u−1) ≡ 0 (mod 3), and ⇌u(u − 1)(g2 − h2) ≡ 0 (mod 12). These necessary conditions are shown to be sufficient for all λ ≥ 2. The known existence result for λ = 1 is also improved.

Optimal Ternary Constant-Composition Codes of Weight Four and Distance Five

Constant-composition codes (CCCs) are a generalization of constant weight codes and permutation codes. The concept of group divisible codes, an analog of group divisible designs in combinatorial design theory, was first introduced by Chee et al. This new class of codes has been shown to be useful in recursive constructions of constant-weight codes and constant-composition codes. In this paper, we consider the problem of determining the maximal sizes of ternary constant-composition codes of weight four and distance five using group divisible codes as the main tools. We determine the exact values for these parameters. The previously known results are those with code length no greater than 10.

Group divisible designs with three unequal groups and larger first index

We show the necessary conditions are sufficient for the existence of group divisible designs (or PBIBDs) with block size k=3 with three groups of size (n,2,1) for any n≥2 and any two indices with λ1>λ2.

The asymptotic existence of group divisible designs of large order with index one

This paper gives the answer to a question of R.M. Wilson regarding the existence of group divisible designs of large order. Let k and u be positive integers such that 2 ⩽ k ⩽ u . Then there exists an integer m 0 = m 0 ( k , u ) such that there exists a group divisible design of group type m u with block size k and index one for any integer m ⩾ m 0 satisfying the necessary arithmetic conditions 1. m ( u − 1 ) ≡ 0 mod ( k − 1 ) , 2. m 2 u ( u − 1 ) ≡ 0 mod k ( k − 1 ) . This paper also presents a large-index asymptotic existence theorem for group divisible t-designs with a fixed number of groups, fixed group size and fixed block size.

Super-simple group divisible designs with block size 4 and index 9

A design is said to be super-simple if the intersection of any two blocks has at most two elements. In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple GDDs are useful in constructing super-simple BIBDs. The existence of super-simple ( 4 , λ ) ‐ GDDs has been determined for λ = 2 – 6 . In this paper, we investigate the existence of a super-simple (4,9)-GDD of group type g u and show that such a design exists if and only if u ≥ 4 , g ( u − 2 ) ≥ 18 and u ( u − 1 ) g 2 ≡ 0 (mod 4).

Super-simple resolvable group divisible designs with block size 4 and index 2

Super-simple resolvable group divisible designs with block size 4 and index 2

Some Optimal Row-Column Designs

Abstract The optimality of two classes of row-column designs with two rows is established. Studies of row-column designs with two rows arise, for example, from designs of two-color microarray experiments. Each mRNA sample to be labeled with green or red fluorescent dyes and hybridized on a slide is a treatment, each slide defines a column, and the green and red dyes define the rows. We show the optimality of designs such that the treatments appear nearly equally often in the two rows, and the column designs are either balanced incomplete block designs or group-divisible designs with two groups and λ2 = λ1 + 1, two well known classes of optimal block designs.

A Complete Generalization of Clatworthy Group Divisible Designs

Partially balanced incomplete block designs (PBIBDs) have a long history and have been extensively used in agriculture and industrial experiments. Since the book of Clatworthy on two-associate-class partially balanced designs was published in 1973, little progress has been made on the construction of these designs. Group divisible designs (GDDs) are an important type of PBIBD with two associate classes. The existence of a GDD with block size $k=3$ was completely settled by Fu, Rodger, and Sarvate. In their works, the most difficult case to solve was when the number of groups, m, is less than the block size k. The existence of GDDs when $m<k$ is, in general, a difficult case to solve. Indeed, when $k=4$, very little is known about the existence of such GDDs. In this paper, we present two general construction methods for GDDs. The first one is a generalization of Wilson's fundamental construction in combinatorial design theory. The second is an extension of the traditional construction using double group divisible designs. As an application of our new construction methods, a complete solution is provided for GDDs that generalize all eleven designs in the old table of Clatworthy which have block size four, three groups, and replication number at most $10$. For these GDDs, no progress has been made until very recently.

Group Divisible Designs with Two Associate Classes and λ2 = 1

The original classiffcation of PBIBDs defined group divisible designs GDD(v = v1 + v2 + ⋯ + vg, g, k, λ1, λ2) with λ1 ≠ 0. In this paper, we prove that the necessary conditions are suffcient for the existence of the group divisible designs with two groups of unequal sizes and block size three with λ2 = 1.

Completely reducible super-simple designs with block size four and related super-simple packings

A design is said to be super-simple if the intersection of any two blocks has at most two elements. A super-simple design $${\mathcal{D}}$$ with point set X, block set $${\mathcal{B}}$$ and index ? is called completely reducible super-simple (CRSS), if its block set $${\mathcal{B}}$$ can be written as $${\mathcal{B}=\bigcup_{i=1}^{\lambda} \mathcal{B}_i}$$ , such that $${\mathcal{B}_i}$$ forms the block set of a design with index unity but having the same parameters as $${\mathcal{D}}$$ for each 1 ? i ? ?. It is easy to see, the existence of CRSS designs with index ? implies that of CRSS designs with index i for 1 ? i ? ?. CRSS designs are closely related to q-ary constant weight codes (CWCs). A (v, 4, q)-CRSS design is just an optimal (v, 6, 4) q+1 code. On the other hand, CRSS group divisible designs (CRSSGDDs) can be used to construct q-ary group divisible codes (GDCs), which have been proved useful in the constructions of q-ary CWCs. In this paper, we mainly investigate the existence of CRSS designs. Three neat results are obtained as follows. Firstly, we determine completely the spectrum for a (v, 4, 3)-CRSS design. As a consequence, a class of new optimal (v, 6, 4)4 codes is obtained. Secondly, we give a general construction for (4, 4)-CRSSGDDs with skew Room frames, and prove that the necessary conditions for the existence of a (4, 2)-CRSSGDD of type g u are also sufficient except definitely for $${(g,u)\in \{(2,4),(3,4),(6,4)\}}$$ . Finally, we consider the related optimal super-simple (v, 4, 2)-packings and show that such designs exist for all v ? 4 except definitely for $${v\in \{4,5,6,9\}}$$ .

On the Existence of Resolvable (K 3 + e)-Group Divisible Designs

In this paper, it is shown that the necessary conditions for the existence of resolvable (K 3 + e)-group divisible designs with group-type g u are also sufficient.

Mixed group divisible designs with three groups and block size 4

A group divisible design (GDD) with three groups and block size 4 is called even, odd, or mixed if the sizes of the non-empty intersections of any of its blocks with any of the three groups are always even, always odd, or always mixed. It has been shown that the necessary conditions for the existence of GDDs of these three types are also sufficient except possibly for the minimal case of mixed designs for group size 5 t ( t > 1 ). In this paper, we complete the undetermined families of mixed GDDs using two constructions based on idempotent self-orthogonal Latin squares and skew Room squares.

Group divisible designs with block size four and group type [formula omitted] where [formula omitted] is a multiple of 8

We determine, up to some possible exceptions, the spectrum for 4-GDDs of type g u m 1 , where g is a multiple of 8 until 48, g is a multiple of 24 until 144, respectively. These spectra are without exceptions for g = 8 , 16, 24, 48, 72, 96, 120 and 144. Furthermore, we establish, up to a finite number of possible exceptions, the spectra for 4-GDDs of types 3 0 u m 1 and 9 0 u m 1 . Finally, we provide nine {3, 5}-URDs which were the last possible exceptions in their classes.

Gregarious GDDs with Two Associate Classes

The existence of group divisible designs with two associate classes has been studied for over 50 years. Probably the most difficult cases to solve are those in which the number of groups is less than the size of the blocks. Recently, such an existence problem was solved in the case where the groups have the same size and the blocks have size 3. In this paper, we continue to focus on blocks of size 3, solving the existence problem when the required designs are gregarious (each block intersects each group). These designs are tight to construct in the sense that they satisfy equality in one of the bounds required for GDDs to exist.

On super-simple group divisible designs with block size four and index

Abstract In this paper, we show that the necessary conditions for the existence of a super-simple ( 4 , λ ) -GDD of type g u with λ ∈ { 3 , 4 , 6 } are also sufficient except for ( λ , g , u ) ∈ { ( 3 , 2 , 5 ) , ( 4 , 3 , 5 ) } .

Two-Dimensional Optical Orthogonal Codes and Semicyclic Group Divisible Designs

A <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n</i> x <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , ¿) two-dimensional optical orthogonal code (2-D OOC), <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> , is a family of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> x <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> (0, 1)-arrays of constant weight <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> such that <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿i</i> =1 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ¿ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</i> =0 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -1 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</i> ) <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</i> ¿ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> ¿) ¿ ¿ for any arrays <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> and any integer <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</i> except when <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿ ¿ 0</i> (mod <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> ), where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿m</i> denotes addition modulo <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> . Such codes are of current practical interest as they enable optical communication at lower chip rate. To simplify practical implementation, the AM-OPPW (at most one-pulse per wavelength) restriction is often appended to a 2-D OOC. An AM-OPPW 2-D OOC is optimal if its size is the largest possible. In this paper, the notion of a perfect AM-OPPW 2-D OOC is proposed, which is an optimal <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n</i> x <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , ¿) AM-OPPW 2-D OOC with cardinality <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[(m</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -1)...( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -¿))/( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -1)...( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -¿))] . A link between optimal <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n</i> x <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , ¿) AM-OPPW 2-D OOCs and block designs is developed. Some new constructions for such optimal codes are described by means of semicyclic group divisible designs. Several new infinite families of perfect <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n</i> x <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , 1) AM-OPPW 2-D OOCs with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> ¿ {2, 3, 4} are thus produced.

Generalizing Clatworthy group divisible designs

In this paper a neat construction is provided for three new families of group divisible designs that generalize some designs from Clatworthy's table of the only 11 designs with two associate classes that have block size four, three groups, and replication numbers at most 10. In each case (namely, λ1=4 and λ2=5, λ1=4 and λ2=2, and λ1=8 and λ2=4), we have proved that the necessary conditions found are also sufficient for the existence of such GDD's with block size four and three groups, with one possible exception.

Super-simple group divisible designs with block size 4 and index 2

Super-simple group divisible designs are useful for constructing other types of super-simple designs which can be applied to codes and designs. In this article, we investigate the existence of a super-simple (4,2)-GDD of type gu and show that such a design exists if and only if u⩾4, g(u−2)⩾4, g(u−1)≡0(mod3) and u(u−1)g2≡0(mod6).

The spectrum of (gυ, g, 3, λ)-dDF in Z gυ

In this paper, several recursive constructions for directed difference family and perfect directed difference family are presented by means of difference matrix and incomplete difference matrix. Finally the necessary and sufficient conditions for the existence of a (gυ, g, 3, λ)-directed difference family in Zgυ are established. As a consequence, the necessary and sufficient conditions for the existence of a cyclic directed group divisible design with block size three and type gυ are obtained.

On uniformly resolvable designs with block sizes 3 and 4

A Uniformly Resolvable Design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k, such a class is denoted k -pc and for a given k the number of k -pcs is denoted r k . In this paper we consider the case of block sizes 3 and 4. The cases r 3 = 1 and r 4 = 1 correspond to Resolvable Group Divisible Designs (RGDD). We prove that if a 4-RGDD of type h u exists then all admissible {3, 4}-URDs with 12hu points exist. In particular, this gives existence for URD with v ? 0 (mod 48) points. We also investigate the case of URDs with a fixed number of k -pc. In particular, we show that URDs with r 3 = 4 exist, and that those with r 3 = 7, 10 exist, with 11 and 12 possible exceptions respectively, this covers all cases with 1 < r 3 ? 10. Furthermore, we prove that URDs with r 4 = 7 exist and that those with r 4 = 9 exist, except when v = 12, 24 and possibly when v = 276. In addition, we prove that there exist 4-RGDDs of types 2 142, 2 346 and 6 54. Finally, we provide four {3,5}-URDs with 105 points.