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

Abstract

In 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.

Partial geometric designs having circulant concurrence matrices

“…(More information on this context and association schemes, we refer the readers to [38] or [1,13].) Theorem 3.15 (Nowak et al [38]). Let  be a 3-class association scheme, and let…”

“…We state the characterization theorem of 3-class association schemes whose relation graphs give rise to PGDs. (More information on this context and association schemes, we refer the readers to [38] or [1,13].) Theorem 3.15 (Nowak et al [38]).…”

“…It is well known that every complete multipartite strongly regular graph gives rise to a PGD. Recently, Nowak, Olmez, and Song [38] have shown that a strongly regular graph with parameters $$({\bf{v}},{\bf{k}},\nu ,\mu )$$ gives rise to a PGD if and only if it satisfies either $${\bf{k}}=\mu $$ or $$\nu =\mu $$. They also have characterized which strongly regular graphs give rise to symmetric PGDs, where symmetric means the number of blocks and the points are the same.…”

Partial geometric designs with block sizes three and four

“…Remark The value of n in the equation (3) is in the definition of a partial geometric design defined in [13,14,18]. …”

“…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]. In this paper, we focus on partial geometric designs with block sizes three and four.…”