Partial Geometric Designs Research Articles

Construction of rank 4 self‐dual association schemes inducing three partial geometric designs

AbstractXu characterized rank 4 self‐dual association schemes inducing three partial geometric designs by their character tables. We construct such association schemes as Schur rings over Abelian 2‐groups.

Self‐dual association schemes, fusions of Hamming schemes, and partial geometric designs

AbstractPartial geometric designs can be constructed from basic relations of association schemes. An infinite family of partial geometric designs were constructed from the fusion schemes of certain Hamming schemes in work by Nowak et al. (2016). A general method to create partial geometric designs from association schemes is given by Xu (2023). In this paper, we continue the research by Xu (2023). We will first study the properties and characterizations of self‐dual association schemes. Then using the characterizations of self‐dual association schemes and the representation theory (character tables) of commutative association schemes, we obtain characterizations and classifications of self‐dual (symmetric or nonsymmetric) association schemes of rank 4 that produce as many as possible nontrivial partial geometric designs or 2‐designs. In particular, for a primitive self‐dual symmetric association scheme of rank 4, if is a power of 3 and each of , , and induces a partial geometric design, then we will prove that is algebraically isomorphic to a fusion scheme of the Hamming scheme for some odd number .

Partial geometric designs arising from association schemes

The basic relations of an association scheme induce block designs in a natural way. There are many (commutative or non-commutative) association schemes in which some basic relations induce partial geometric designs or 2-designs. In this paper we study how the algebraic structures of association schemes determine partial geometric designs. We will first establish a necessary and sufficient condition under which some basic relations of an association scheme induce partial geometric designs. Then by using the representation theory of association schemes, we will show that the parameters of these partial geometric designs have some special properties. We will also use the characters of commutative association schemes to determine which basic relations induce partial geometric designs or 2-designs. Isomorphisms of partial geometric designs will also be discussed.

Partial geometric designs having circulant concurrence matrices

AbstractWe classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the concurrence matrix of a PGD can have at most three distinct eigenvalues, all of which are nonnegative integers. The matrix contains useful information on the incidence structure of the design. An ordinary 2‐ design has a single concurrence , and its concurrence matrix is circulant, a partial geometry has two concurrences 1 and 0, and a transversal design has two concurrences and 0, and its concurrence matrix is circulant. In this paper, we survey the known PGDs by highlighting their concurrences and constructions. Then we investigate which symmetric circulant matrices are realized as the concurrence matrices of PGDs. In particular, we try to give a list of all PGDs of order up to 12 each of which has a circulant concurrence matrix. We then describe these designs along with their combinatorial properties and constructions. This work is part of the second author's Ph.D. dissertation [46].

Plateaued functions on finite abelian groups and partial geometric difference sets

AbstractAs a generalization of plateaued functions on finite fields and bent functions (perfect nonlinear functions) on finite abelian groups, plateaued functions on finite abelian groups were introduced in [B. Xu, Plateaued functions, partial geometric difference sets, and partial geometric designs, J. Combin. Des. 27 (2019), 756–783]. In this paper, we continue the research in the paper mentioned above. We will obtain various characterizations of plateaued functions; these characterizations establish close connections between plateaued functions and some combinatorial objects: partial geometric difference sets and related partial geometric difference families. Then we introduce the complementary matrix and Cayley matrix for a subset of a finite group and use them to characterize partial geometric difference sets. As applications, we will show how to construct directed strongly regular graphs from partial geometric difference sets and establish a natural relation between partial geometric difference sets and partial geometric designs. The tensor product of a group algebra and a cyclotomic field is an important tool for our discussions.

Partial geometric designs with block sizes three and four

AbstractIn this paper, we focus on partial geometric designs with block sizes three and four. First, we show that a partial geometric design with block size three is a 2‐design, or a transversal design, or each line of a generalized quadrangle that is repeated times. Second, we introduce the concept of type of partial geometric designs with block size four and characterize partial geometric designs of type for some integer .

Three-class association schemes from partial geometric designs

In this paper, we show that three-class association schemes can be constructed from certain partial geometric designs. First, we show that partial geometric designs with exactly three indices: r−n, λ1 and λ2 give rise to three-class association schemes, using the method of Beker and Haemers (1980). Second, we describe parameter sets of certain partial geometric designs that also give rise to three-class association schemes.

Plateaued functions, partial geometric difference sets, and partial geometric designs

AbstractPlateaued functions on finite fields have been studied in many papers in recent years. As a generalization of plateaued functions on finite fields, we introduce the notion of a plateaued function on a finite abelian group. We will give a characterization of a plateaued function in terms of an equation of the matrix associated to the function. Then we establish a one‐to‐one correspondence between the ‐valued plateaued functions and partial geometric difference sets (with specific parameters) in finite abelian groups. We will also discuss two general methods (extension and lifting) for the construction of new partial geometric difference sets from old ones in (abelian or nonabelian) finite groups, and construct many partial geometric difference sets and plateaued functions. A one‐to‐one correspondence between partial geometric difference sets (in arbitrary finite groups) and partial geometric designs will be proved.

Partial geometric designs from group actions

In this paper, using group actions, we introduce a new method for constructing partial geometric designs (sometimes referred to as $1\frac{1}{2}$-designs). Using this new method, we construct several infinite families of partial geometric designs by investigating the actions of various linear groups of degree two on certain subsets of $\mathbb{F}_{q}^{2}$. Moreover, by computing the stabilizers of such subsets in various linear groups of degree two, we are also able to construct a new infinite family of balanced incomplete block designs.

Partial geometric difference sets and partial geometric difference families

The concept of a partial geometric difference set (or 112-difference set) was introduced by Olmez in 2014. Recently, Nowak, Olmez and Song introduced the notion of a partial geometric difference family, which generalizes both the classical difference family and the partial geometric difference set. It was shown that partial geometric difference sets and partial difference families give rise to partial geometric designs. In this paper, a number of new infinite families of partial geometric difference sets and partial geometric difference families are constructed. From these partial geometric difference sets and difference families, we generate a list of infinite families of partial geometric designs.

Constructions of 112-Designs from Unitary Geometry over Finite Fields

In this paper, we construct some [Formula: see text]-designs, which are also known as partial geometric designs, using totally isotropic subspaces of the unitary space. Furthermore, these [Formula: see text]-designs yield six infinite families of directed strongly regular graphs.

A framework for constructing partial geometric difference sets

Partial geometric difference sets (PGDSs) were defined in Olmez (J Combin Des 22(6):252–269, 2014). They are used to construct partial geometric designs. We use the framework of extended building sets to find infinite families of PGDSs in abelian groups. Included in our new families of PGDSs are generalizations of the Hadamard, McFarland, Spence, Davis-Jedwab, and Chen difference sets.

A link between combinatorial designs and three-weight linear codes

In this paper, we show that partial geometric designs can be constructed from certain three-weight linear codes, almost bent functions and ternary weakly regular bent functions. In particular, we show that existence of a family of partial geometric difference sets is equivalent to existence of a certain family of three-weight linear codes. We also provide a link between ternary weakly regular bent functions, three-weight linear codes and partial geometric difference sets.

New partial geometric difference sets and partial geometric difference families

Using Galois rings and Galois fields, we construct several infinite classes of partial geometric difference sets, and partial geometric difference families, with new parameters. Furthermore, these partial geometric difference sets (and partial geometric difference families) correspond to new infinite families of directed strongly regular graphs. We also discuss some of the links between partially balanced designs, 2-adesigns (which were recently coined by Cunsheng Ding in “Codes from Difference Sets” (2015)), and partial geometric designs, and make an investigation into when a 2-adesign is a partial geometric design.

A Family of Partial Geometric Designs from Three‐Class Association Schemes

AbstractIn this paper, we show that partial geometric designs can be constructed from certain three‐class association schemes and ternary linear codes with dual distance three. In particular, we obtain a family of partial geometric designs from the three‐class association schemes introduced by Kageyama, Saha, and Das in their article [“Reduction of the number of associate classes of hypercubic association schemes,” Ann Inst Statist Math 30 (1978)]. We also give a list of directed strongly regular graphs arising from the partial geometric designs obtained in this paper.

Constructions of [formula omitted]-designs from orthogonal geometry over finite fields

In this paper, we construct some 112-designs, which are also known as partial geometric designs, using totally isotropic subspaces of orthogonal spaces. Furthermore, these 112-designs yield some infinite families of directed strongly regular graphs.

Constructions of 1 1/2-designs from symplectic geometry over finite fields

In this paper, we construct some 1 1/2-designs, which are also known as partial geometric designs, using totally isotropic subspaces of the symplectic space and generalized symplectic graphs. Furthermore, these 1 1/2-designs yield six infinite families of directed strongly regular graphs.

Partial geometric designs with prescribed automorphisms

Combinatorial designs have long been used to design efficient statistical experiments. More recently, connections to the theory of cryptographic communications have emerged. Combinatorial designs have provided solutions to problems coming from signal processing, radar, error-correcting codes, optical orthogonal codes, and image processing. Further, the most elegant solutions have come from designs with prescribed automorphisms. In this paper, we focus on partial geometric designs, a generalization of the classical 2-design. These designs have recently been shown to produce two directed strongly regular graphs. We generalize the well-known Kramer---Mesner Theorem for 2-designs to partial geometric designs. We also construct infinite families of partial geometric designs admitting a group of automorphisms which acts regularly on the point set and semi-regularly on the block set. The designs are obtained from new constructions of partial geometric difference families. These families were recently introduced a generalization of both the classical difference family and the partial geometric difference set.

Partial Geometric Difference Families

AbstractWe introduce the notion of a partial geometric difference family as a variation on the classical difference family and a generalization of partial geometric difference sets. We study the relationship between partial geometric difference families and both partial geometric designs and difference families, and show that partial geometric difference families give rise to partial geometric designs. We construct several infinite families of partial geometric difference families using Galois rings and the cyclotomy of Galois fields. From these partial geometric difference families, we generate a list of infinite families of partial geometric designs and directed strongly regular graphs.

Symmetric ‐Designs and ‐Difference Sets

AbstractIn this paper, we introduce a method to construct ‐designs, which are also known as partial geometric designs, by using subsets of certain finite groups. We introduce the concept of ‐difference sets and investigate the existence and nonexistence of these structures. We also provide some nonexistence results on ‐designs based on the fact that ‐designs yield directed strongly regular graphs.