pub fn association_scheme_multiplicities(
num_vars: usize,
clauses: &[Vec<Lit>],
) -> Option<Vec<u128>>Expand description
The multiplicities of the association scheme — the dimensions m_j of the common eigenspaces of the
relation matrices in the full |X|-dimensional space; the “degrees” of the scheme, dual to the valencies.
From the eigenmatrix P (#eigenmatrix) and the scheme orthogonality relation
m_j = |X| / Σ_i P[j][i]·P[j][ī]/k_i (k_i = valency, ī = transpose relation). Computed exactly over a
GF(p) chosen to split the algebra AND exceed |X| (so the small positive integer m_j ≤ |X| decodes
uniquely). Returns the multiplicities sorted ascending.
FAIL-CLOSED: None unless every m_j is a positive integer ≤ |X|, Σ_j m_j = |X| (the eigenspaces
partition the space), and the trivial eigenspace (the valency row of P) has multiplicity 1. For a
multiplicity-free group action these are exactly the degrees of the constituent irreducibles.