Research proof index

Folding preserves distance from Reed–Solomon codes. For any word f over the smooth coset FFT domain, degree parameter d, folding parameter k, and distance threshold δ satisfying 0 < δ < min (δᵣ(f, RS[d])) (1 - sqrtRate(d)), the probability over a uniformly random folding challenge r : F that the folded word is within relative distance δ of t...

An explicit formula for foldWord when k = 1 that does not use Lagrange interpolation and avoids using log.

Perfect completeness of folding: if a word belongs to an RS-code then its foldWord belongs to a folded RS-code.

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).