Equidistant Codes in the Grassmannian

TL;DR

This paper explores equidistant codes in the Grassmannian, introducing new constructions including a Plücker embedding-based method for 1-intersecting codes and related constant rank codes, advancing understanding of their maximal sizes and structures.

Contribution

The paper presents novel constructions for equidistant codes in the Grassmannian, including a new method based on the Plücker embedding for 1-intersecting codes and related constant rank codes.

Findings

Largest codes are sunflowers for large vector spaces.

New construction achieves specific code sizes using Plücker embedding.

Related constant rank codes with high rank distance are constructed.

Abstract

Equidistant codes over vector spaces are considered. For $k$-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Pl\"{u}cker embedding, for 1-intersecting codes of $k$-dimensional subspaces over $\F_q^n$, $n \geq \binom{k+1}{2}$, where the code size is $\frac{q^{k+1}-1}{q-1}$ is presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size $n \times \binom{n}{2}$ over $\F_q$, rank $n-1$, and rank distance $n-1$.