Combinatorial constructions of Schubert subspace codes

2026-07-08

AI Summary

This study addresses the construction of very large constant-dimension subspace codes—specifically Schubert subspace codes—under constraints imposed by intersection with a fixed subspace. Two novel construction methods are proposed: the first integrates a direct-sum decomposition of the ambient space, partial spreads, and colorings of the $q$-Johnson graph; the second leverages field-reduction techniques to derive codewords from avoidance-type and scattered-type subspaces defined over extension fields. The work establishes fundamental bounds on code size dictated by the chromatic and clique numbers of the $q$-Johnson graph, thereby unifying and generalizing the only previously known construction. In the extremal-distance regime, the constructed codes attain the natural counting upper bound, while in the scattered case, their cardinality can be determined exactly.

Abstract

We study Schubert subspace codes, which are constant-dimension subspace codes with prescribed intersection conditions with a fixed subspace. Our goal is to construct codes of maximum possible size in the extremal distance cases where a natural counting upper bound applies. We give two families of constructions. The first one uses a direct-sum decomposition of the ambient space, together with partial spreads and colorings of powers of $q$-Johnson graphs. For this construction, we also prove necessary conditions, which show how chromatic and clique obstructions arise. The second family is obtained by field reduction from evasive and scattered subspaces over extension fields. This gives codes whose size can be computed exactly in the scattered case and recovers the only previously known construction as a special case.

Problem

• Schubert subspace codes
• constant-dimension subspace codes
• intersection conditions
• extremal distance
• maximum size

Innovation

• Schubert subspace codes
• partial spreads
• q-Johnson graphs
• field reduction
• scattered subspaces