Exact bounds on the sizes of covering codes (Q1406876)

The main goal of this paper is to show how the results of extremal hypergraph theory can be applied to find connections between Turán theory and constant weight covering codes. In particular, for \(n> n_0(r)\), the authors give the exact minimum number of Hamming balls of radius \(r\) required to cover a Hamming ball of radius \(r+2\) in \(\{0,1\}^n\). It is proved that \[ c_r(B_n(0, r+ 2))= \sum_{1\leq i\leq r+1} \begin{pmatrix} \lfloor(n+ i-1)/(r+1)\rfloor\\ 2\end{pmatrix}+ \left\lfloor {n\over r+1}\right\rfloor \] and that the centers of the covering balls \(B(x,r)\) can be obtained by taking all pairs in the parts of an \((r+1)\)-partition of the \(n\)-set and by taking the singletons in one of the parts.