Bounds on Subspace Codes Based on Subspaces of Type (m,1) in Singular Linear Space

Abstract

The Sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on the subspace codes ${\left(n+l,M,d,(m,1)\right)}_q$ based on subspaces of type $(m,1)$ in singular linear space ${\mathbb{F}}_q^{(n+l)}$ over finite fields ${\mathbb{F}}_q$ are presented. Then, we prove that codes based on subspaces of type $(m,1)$ in singular linear space attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in ${\mathbb{F}}_q^{(n+l)}$.