Bounds on the cardinality of subspace codes with non-maximum code distance (Q2068654)

This paper studies subspace codes with nonmaximum code distance. The subspace distance between two subspaces \(U, V \in \mathrm{GF}(q)^n\) is \[d_{\text{sub}}(U,V) = \dim(U) + \dim(V) -2\dim(U\cap V ).\] If \(U, V\) are of the same dimension \(m,\) the subspace distance equals \(d_{\text{sub}}(U, V)=2(m-\dim(U\cap V)) = 2\delta\), \(\delta=m-\dim(U\cap V)\) known as the Grassmannian metric. Families of nonspreads based on using the Silva-Kotter-Kschischang (SKK) subspace code construction and Gabidulin-Bossert multicomponent codes with zero prefix (MZP) are considered. Numerical estimates for cardinalities of nonspreads for a large number of parameters are given. Moreover, it is shown that for large dimensions all the three codes almost attain the maximum cardinality bound given by the Johnson inequality \[M(n + 1,d_{\text{sub}},m + 1)\leq \frac{q^{n+1}-1}{q^{m+1}-1}M(n, d_{\text{sub}}, m).\] The authors present examples of nonspreads with the best values of the cardinality against previously known subspace codes with the same parameters.