An Asymptotic Formula for the Number of Balanced Incomplete Block Design Incidence Matrices

TL;DR

This paper establishes an asymptotic formula for counting incidence matrices of balanced incomplete block designs by linking them to a specific random walk on a Euclidean lattice and analyzing its return probability.

Contribution

It introduces a novel connection between BIBD incidence matrices and a Euclidean lattice random walk, enabling asymptotic enumeration.

Findings

Derived the return probability of the random walk.

Obtained an asymptotic formula for the number of BIBD incidence matrices.

Extended methods similar to those used for partial Hadamard matrices.

Abstract

We identify a relationship between a random walk on a certain Euclidean lattice and incidence matrices of balanced incomplete block designs. We then compute the return probability of the random walk and use it to obtain the asymptotic number of BIBD incidence matrices (as the number of columns increases). Our strategy is similar in spirit to the one used by de Launey and Levin to count partial Hadamard matrices.