  Codes in finite projective spaces equipped with the subspace distance have
been proposed for error control in random linear network coding. The resulting
so-called \emph{Main Problem of Subspace Coding} is to determine the maximum
size $A_q(v,d)$ of a code in $\operatorname{PG}(v-1,\mathbb{F}_q)$ with minimum
subspace distance $d$. Here we completely resolve this problem for $d\ge v-1$.
For $d=v-2$ we present some improved bounds and determine $A_q(5,3)=2q^3+2$
(all $q$), $A_2(7,5)=34$. We also provide an exposition of the known
determination of $A_q(v,2)$, and a table with exact results and bounds for the
numbers $A_2(v,d)$, $v\leq 7$.
