Research proof index

Generalisation of jointAgreement_implies_second_proximity to an arbitrary word stack over a submodule code. If a stack W : Fin k → ι → F jointly agrees with a submodule C ⊆ ι → F, then every element of the linear span of the stack is δ-close to C. The pointwise case W i ∈ C is the special case x = W i (choose coefficients c to be the i-th...

The coefficient-parameterised probability equals the subtype probability. The map r ↦ u₀ + ∑ rᵢ • dᵢ has constant-cardinality fibers (cosets of the kernel of the linear part), so pushforward of uniform gives uniform.

Relative distance to a submodule is invariant under scaling by a unit: δᵣ(z • u, V) = δᵣ(u, V) for z ≠ 0 and V a submodule. Key step in BCIKS20 §6.3 (Step 1c).

Lemma 4.4, [AHIV22] (mutual-exclusion corollary). Either all points on the affine line are e-close to the Reed–Solomon code, or at most ‖RS‖₀ points are. The assumptions v ≠ 0 and ‖RS‖₀ < |F| are necessary for mutual exclusion: if v = 0, the affine line degenerates to a singleton and the two branches can hold simultaneously.

If A and B are coprime and agree on sufficiently many lines, then A is constant.

Main existence: if B/A agrees with low-degree quotients on enough lines, then A ∣ B.