On an infinite sequence of strongly regular digraphs with parameters $(9(2n+3), 3(2n+3), 2n+4, 2n+1, 2n+4)$

Abstract

The paper constructs an infinite sequence of strongly regular directed graphs. The construction is based on representing adjacency matrices as block matrices composed of circulant blocks, together with the use of a compactification operation consistent with polynomial arithmetic modulo $x^{2n+3}-1$. Using computer search with the pychoco library and subsequent analysis of automorphism groups in the GAP system, a stable structural pattern was identified, which made it possible to formulate and prove an explicit formula for the adjacency matrices of the infinite sequence of directed graphs. Among the obtained digraphs, there are examples with parameters $(63, 21, 8, 5, 8)$ and $(81, 27, 10, 7, 10)$, for which the question of existence had previously remained open. A hypothesis on the structure of the automorphism groups of the digraphs in the constructed sequence is also formulated.