Schubert Subspace Codes
Gianira N. Alfarano, Joachim Rosenthal, Beatrice Toesca
TL;DR
This work introduces Schubert subspace codes, constant-dimension subspace codes restricted to Schubert varieties, and an intersecting-sets perspective with respect to a fixed subspace. It develops maximal-size constructions for minimum subspace distance $d_S=2k$ in regimes where $n=rk$ and $u\le\frac{rk}{2}$, leveraging linear sets and field reduction, and it compares these with the Etzion–Silberstein multilevel construction, showing regimes where the new approach yields larger codes. The paper then generalizes to a broader $(\ell,t)$-intersecting framework, providing lower bounds via Ferrers-diagram codes and multilevel constructions and proving simple upper bounds in certain parameter regimes. Overall, the results connect Schubert calculus, linear sets, and rank-metric/Ferrers-diagram code techniques to advance structured, high-distance constant-dimension codes and pose several open questions about extensions and tight bounds. The work thus offers new geometric tools and constructions with potential impact on network coding and related combinatorial coding theory contexts.
Abstract
In this paper, we initiate the study of constant dimension subspace codes restricted to Schubert varieties, which we call Schubert subspace codes. These codes have a very natural geometric description, as objects that we call intersecting sets with respect to a fixed subspace. We provide a geometric construction of maximum size constant dimension subspace codes in some Schubert varieties with the largest possible value for the minimum subspace distance. Finally, we generalize the problem to different values of the minimum distance.