Low-rank-free subcodes in random matrix codes

Determine whether a uniformly random matrix code of the relevant dimension contains a subcode of codimension $\ell_s$ with no nonzero matrices of rank at most $\kappa$, as required to compare the MIRANDA dual public code with random codes.

Background

For MIRANDA, the paper proves that the dual public code contains a subcode of codimension ℓs\ell_s whose nonzero matrices all have rank greater than κ\kappa. A second-moment argument shows that a random code contains a low-rank matrix with overwhelming probability, but a union bound over all codimension-ℓs\ell_s subcodes is too weak to decide whether some such subcode can nevertheless avoid all matrices of rank at most κ\kappa. Consequently, the distinguishing property remains unresolved.

References

Whether a random code of dimension $K$ has a subcode of codimension $\ell_s$ without nonzero matrices of rank at most $\kappa$ is not decided by this argument: a union bound over its subcodes of codimension $\ell_s$ multiplies the above probability by $\genfrac{[}{]}{0pt}{2}{K}{\ell_s}_q\approx q{\ell_s(K-\ell_s)}$, which is too large here. We leave this question open.

— Code distances of matrix rank-metric codes under Add-and-Remove transformations  (2610.03460 - Alfarano et al., 2 Oct 2026) in Section 4, Application to MIRANDA and an Example