Coset Construction for Subspace Codes

TL;DR

This paper introduces a generalized coset construction method for subspace codes in network coding, improving bounds and achieving the MRD bound for many parameters.

Contribution

It extends the coset construction to a broader set of parameters, enhancing the size of subspace codes and reaching the MRD bound in an infinite parameter family.

Findings

Improves known bounds for subspace code sizes.

Attains the MRD bound for an infinite family of parameters.

Generalizes the coset construction for broader applicability.

Abstract

One of the main problems of the research area of network coding is to compute good lower and upper bounds of the achievable cardinality of so-called subspace codes in $\operatorname{PG}(n,q)$, i.e., the set of subspaces of $\mathbb{F}_q^n$, for a given minimal distance. Here we generalize a construction of Etzion and Silberstein to a wide range of parameters. This construction, named coset construction, improves or attains several of the previously best-known subspace code sizes and attains the MRD bound for an infinite family of parameters.