Dimension of subfield subcodes in the base LERS construction

Determine the dimension of the subfield subcodes arising in the randomized linear exact repair scheme for a general linear code, in order to establish whether the rank condition for constructing the repair scheme is satisfied beyond the currently analyzed parameter range.

Background

The paper reduces the construction of leakage functions to a rank condition involving a subfield subcode associated with the punctured dual code. A lower bound shows that the condition is satisfied with overwhelming probability when the base-code dimension obeys a particular bound, but the argument does not provide a general characterization of the relevant subfield-subcode dimension.

The unresolved issue is important because controlling this dimension would determine when the randomized construction yields a linear exact repair scheme for Massey secret sharing schemes induced by arbitrary linear codes.

References

However, analyzing this rank is difficult, since it requires controlling the dimension of subfield subcodes, which is not known in general.

— On the Leakage of Massey Secret Sharing Schemes under Linear Computations  (2609.19929 - Aoutouf et al., 17 Sep 2026) in Section 2.5, “Randomized Method”