Precise conditions for dual–star product equality in subfield-subcodes of J-affine variety codes

Identify the precise conditions under which the equality S(C_1 \star C_2)^\perp = S(C_1)^\perp \star S(C_2)^\perp holds for subfield-subcodes of J-affine variety codes, where S(\cdot) denotes the subfield-subcode operator and \star denotes the componentwise (Schur) product. Establish a complete characterization of when the dual of the subfield-subcode of the star product coincides with the star product of the dual subfield-subcodes, to clarify the interaction between duality and Schur products in this setting.

Background

The paper studies Schur products of evaluation codes and applications to CSS–T quantum codes and PIR. For secure multi-party computation, one needs to relate the componentwise squares of codes and their duals. The authors show via a counterexample (Remark le:SSDual) that, in general, the dual of the subfield-subcode of a star product does not equal the star product of the dual subfield-subcodes.

To enable stronger algebraic guarantees in cryptographic applications, it is necessary to know exactly when the equality S(C_1 \star C_2)^\perp = S(C_1)^\perp \star S(C_2)^\perp holds. The authors explicitly defer determining these conditions, marking it as future work.