Research proof index

For a single coordinate (ℓ = 1), membership in the special-sound family SS(S, 1, k) is exactly injectivity of the k challenge vectors: a central vector together with k-1 siblings differing in the unique coordinate is just k pairwise-distinct values. This is the cleaner ℓ = 1 characterization underlying the bridge to plain k-special soundness.

Generic single-round CWSS assembly. Any pure statement-extending verifier of the two-round pSpec is coordinate-wise special sound for foldStructure, provided a witness assembler mkWitness that turns per-branch relOut-witnesses at star-shaped challenge families into a relIn-witness. This discharges all tree/extractor plumbing once; the protoc...

The tensor product generator TensorGenerator and the explicit componentwise generator TensorGenerator_Explicit agree under the canonical isomorphism between F^ℓ ⊗ F^ℓ′ and (ℓ × ℓ') → F.

A linear code LC of length ι and dimension dim over a field F is MDS if any dim columns of the generator matrix whose rows are an F-basis of LC are linearly independent.[GRS25] Equivalently, a linear code is MDS if and only if its generator matrix is MDS. Note: the hypothesis 0 < dim LC is necessary because for a trivial code (dim = 0),...

If a generator matrix is MDS with at least one row, then the code it generates is MDS.

If a linear code generated by a full-rank matrix is MDS, then the matrix is MDS.