Frequency of exceptional subcode-distance decreases

Determine how often the decrease in subcode distance occurs at the exceptional indices of Add-and-Remove codes, particularly for the parameter sets used by MIRANDA.

Background

For a matrix Gabidulin code subjected to the Add-and-Remove construction, the paper proves that subcode distances agree with those of the original code except possibly at the last removed indices in each block of indices of length equal to the larger matrix dimension. At those exceptional indices, the distance can decrease by one, and an example demonstrates that such a decrease can occur. The frequency of this phenomenon, especially under MIRANDA's concrete parameters, is not determined.

References

Several questions remain open: how often the decrease at the exceptional indices occurs, in particular for the $MIRANDA$ parameters, what happens for MRD codes that are not Gabidulin codes, and whether subcode distances can be used in a polynomial-time distinguisher.

— Code distances of matrix rank-metric codes under Add-and-Remove transformations  (2610.03460 - Alfarano et al., 2 Oct 2026) in Section 1, Introduction