Partial geometric designs with prescribed automorphisms

Abstract

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.

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

“…Then N is the incidence matrix of a 1‐ design if and only if and ; a 2‐ design if and only if , and a symmetric 2‐ design if and only if , where denotes the transpose of N , and I and J are the identity and the all‐ones matrix, respectively. The incidence matrix of a partial geometric design is a ‐matrix N such that, for certain constants , we have For more information on partial geometric designs we refer the readers to or .…”

“…For more information on partial geometric designs we refer the readers to [21] or [11,22,23,[26][27][28]].…”

“…have recently shown that partial geometric designs give rise to directed strongly regular graphs. In , the authors of the current paper have uncovered which difference sets and difference families produce partial geometric designs. Here, we take the next step and explore the link between partial geometric designs and other combinatorial structures.…”

Partial geometric designs with block sizes three and four

“…In [17], Olmez showed how partial geometric designs can be used to construct plateaued functions. In [18], Olmez investigated the link between partial geometric designs and three‐weight codes, and in [13] Nowak and Olmez described a method that can often be used to determine the existence or nonexistence of a partial geometric design having specified automorphisms. Many researchers constructed partial geometric designs with various objects [7–9,11,14].…”

Partial geometric designs from group actions

“…Brouwer, Olmez and Song, in [5], showed that directed strongly regular graphs can be constructed from partial geometric designs. In [28], Olmez showed how partial geometric designs can be used to construct plateaued functions, in [29], Olmez investigated the link between partial geometric designs and three-weight codes, and in [24], Nowak and Olmez constructed partial geometric designs with prescribed automorphisms.…”