Lower bounds for q-ary codes of covering radius one Original Research Article

Let kq(n) denote the minimal cardinality of a q-ary code C of length n and covering radius one. The numbers of elements of C that lie in a fixed k-dimensional subspace of {0,…,q−1}n satisfy a certain system of linear inequalities. By employing a technique for dealing with ‘large’ values of k (i.e. unbounded with increasing n) we are able to derive lower bounds for kq(n). The method works especially well in cases where the sphere covering bound has not been substantially improved, for example if q=3 and n≡1 (mod 3). As an application we show that the difference between kq(n) and the sphere covering bound approaches infinity with increasing n if q is fixed and (q−1)n+1 does not divide qn. Moreover, we present improvements of already known lower bounds for kq(n) such as k3(10)⩾2835.