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

The Sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on the subspace codes n+l,M,d,(m,1)q based on subspaces of type (m,1) in singular linear space Fq(n+l) over finite fields Fq 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 Fq(n+l)