A nonconstructive upper bound on covering radius

Let
t(n,k)
denote the minimum covering radius of a binary linear
(n,k)
code. We give a nonconstructive upper bound on
t(n,k)
, which coincides asymptotically with the known lower bound, namely
n^{-1}t(n,nR)=H^{-1}(1-R)+O(n^{-l}\log n)
, where
R
is fixed,
0<R<1
, and
H^{-1}
is the inverse of the binary entropy function.