On the covering radius of codes

The covering radius
R
of a code is the maximal distance of any vector from the code. This work gives a number of new results concerning
t[n, k]
, the minimal covering radius of any binary code of length
n
and dimension
k
. For example
t[n, 4]
and
t[n, 5]
are determined exactly, and reasonably tight bounds on
t[n, k]
are obtained for any
k
when
n
is large. These results are found by using several new constructions for codes with small covering radius. One construction, the amalgamated direct sum, involves a quantity called the norm of a code. Codes with norm
\leq 2 R + 1
are called normal, and may be combined efficiently. The paper concludes with a table giving bounds on
t[n, k]
for
n \leq 64
.