A lower bound for constant dimension codes from multi-component lifted MRD codes

In this work we investigate unions of lifted MRD codes of a fixed dimension and minimum distance and derive an explicit formula for the cardinality of such codes. This will then imply a lower bound on the cardinality of constant dimension codes. We will first repeat some known results needed in this section. In Section 2 we will explain the construction and derive the formula for the cardinality and the lower bound. We conclude in Section 3. Let Fq be the finite field with q elements. Gq(k, n) denotes the set of all k-dimensional vector spaces over Fq , called the Grassmannian. A constant dimension code is simply a subset of Gq(k, n). A metric on Gq(k, n) is the injection metric, given by