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

2014/07/17 by Aaron Montgomery, Aaron M. Montgomery, Montgomery, Aaron M.
Biochemistry, Genetics and Molecular Biology · Computer Science · Engineering · Mathematics · #05B20 (Primary) 60J10 (Secondary) #Coding theory and cryptography #Combinatorics (math.CO) #DNA and Biological Computing #FOS: Mathematics #Probability (math.PR) #graph theory and CDMA systems #math.CO #math.PR #msc:05B20 #msc:60J10

paper · pdf · doi:10.48550/arxiv.1407.4552

32 pages; this version corrects typographical errors and includes some clarifications

openalex publication_date 2014/07/17 · arxiv created 2016/02/17 · arxiv updated 2016/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• A Fourier-analytic Approach to Counting Partial Hadamard Matrices
• Mutual incidence matrix of two balanced incomplete block designs
• Asymptotics for incidence matrix classes
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