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Encoding a fixed-size incidence subset with binary wires

For a fixed array, pair (g,h) has C_gh=binom(N_gh,d_gh) possible subsets. Here binom(N,d)=N!/(d!(N-d)!) counts size-d subsets, with 0!=1. Join test-group vertex g to variable-group vertex h with b_gh=ceil(log2 C_gh) ordered parallel wires. Their bits index zero through 2^b_gh-1; both endpoint tensors decode the SAME index. Values at least C_gh are invalid and give entry zero. If C_gh=1, use no wires and take the unique subset, even when it is the entire incidence set. For N_gh=3,d_gh=1, two wires encode three singleton subsets; the fourth bit string must not add an extra contribution.

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Updated 2026-11-08

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Math