Quadratic Checks in the Grassmann Encoding

Determine how to incorporate checks of quadratic equations on an encoded message into the Grassmann encoding while preserving the required properties for constructing d-to-1 games.

Background

The paper considers replacing linear equations with quadratic equations to obtain perfect completeness. The standard Grassmann encoding supports linear checks, but the outer PCP in the perfect-completeness approach requires quadratic checks.

The authors introduce a quadratic Grassmann encoding as a possible workaround, but identify unresolved codeword-testing and covering-property failures. In particular, they explicitly state that they do not know how to incorporate quadratic equation checks directly into the Grassmann encoding.

References

The Grassmann encoding is compatible with the former type of checks, but not with the latter type of checks, in the sense that we do not know how to incorporate checking quadratic equations on the encoded message into the Grassmann encoding itself.

— On the Hardness of $4$-to-$1$ Games with Perfect Completeness  (2610.12378 - Fei et al., 8 Oct 2026) in Section 2.3.1, "A First Attempt: the Quadratic Grassmann Encoding"