On the Probability Distribution of Superimposed Random Codes

A systematic study of the probability distribution of superimposed random codes is presented through the use of generating functions. Special attention is paid to the cases of either uniformly distributed but not necessarily independent or non uniform but independent bit structures. Recommendations for optimal coding strategies are derived.

💡 Research Summary

The paper investigates the probabilistic behavior of superimposed random coding, a technique widely used to accelerate subgraph searches in large chemical‑structure databases. Each compound is represented by a binary descriptor vector; a query is similarly encoded, and the final “target” bitstring is obtained by OR‑combining the codewords associated with the positions of 1‑bits in the original descriptor. This operation must preserve the partial order of the source vectors, which leads to the defining equation ψ(β)=∨_{j∈β^{-1}(1)}ψ_j.

The authors introduce isotropic probability distributions on the n‑bit space, i.e., distributions that depend only on the number of 1‑bits, not on their positions. Such a distribution is characterized by probabilities p_k (k = 0…n) and its generating function f(t)=∑_{k=0}^n C(n,k)p_k t^k. From f(t) they define two auxiliary functions: F_a = P(ξ ≤ α) and G_a = P(ξ ≥ α), which are related to f(t) by simple transformations (F(t)=(1+t)^n f(t/(1+t)), G(t)=t^n f(1/(1+t))).

The central theoretical contribution (Theorem 1) states that if the source bit patterns have generating function Π(t) and the codewords have distribution coefficients F_m, then the distribution of the target bits is given by \hat F_m = Π(F_m). In other words, the source distribution acts as a linear operator on the code‑word distribution, while the code‑word distribution itself introduces a non‑linear transformation. This relationship enables closed‑form expressions for the first two moments of the target distribution:

\hat μ_1 = n