Optimal subspace codes in ${\rm PG}(4,q)$

TL;DR

This paper explores optimal subspace codes in projective 4-space, providing examples of mixed-dimension codes and establishing a key size result for certain parameters.

Contribution

It introduces new optimal mixed-dimension subspace codes in PG(4,q) and determines the maximum size for specific code parameters.

Findings

Constructed examples of optimal mixed-dimension subspace codes

Established that ${ m A}_q(5,3) = 2(q^3+1)$

Demonstrated the existence of codes with non-constant dimension

Abstract

We investigate subspace codes whose codewords are subspaces of ${\rm PG}(4,q)$ having non-constant dimension. In particular, examples of optimal mixed-dimension subspace codes are provided, showing that ${\cal A}_q(5,3) = 2(q^3+1)$.