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We identify a relationship between a random walk on a certain Euclideanlattice and incidence matrices of balanced incomplete block designs. We thencompute the return probability of the random walk and use it to obtain theasymptotic number of BIBD incidence matrices (as the number of columnsincreases). Our strategy is similar in spirit to the one used by de Launey andLevin to count partial Hadamard matrices.

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