On Asymmetric Coverings and Covering Numbers
<|reference_start|>On Asymmetric Coverings and Covering Numbers: An asymmetric covering D(n,R) is a collection of special subsets S of an n-set such that every subset T of the n-set is contained in at least one special S with |S| - |T| <= R. In this paper we compute the smallest size of any D(n,1) for n <= 8. We also investigate ``continuous'' and ``banded'' versions of the problem. The latter involves the classical covering numbers C(n,k,k-1), and we determine the following new values: C(10,5,4) = 51, C(11,7,6,) =84, C(12,8,7) = 126, C(13,9,8)= 185 and C(14,10,9) = 259. We also find the number of nonisomorphic minimal covering designs in several cases.<|reference_end|>
@article{applegate2002on,
    title={On Asymmetric Coverings and Covering Numbers},
    author={David Applegate, E. M. Rains and N. J. A. Sloane},
    journal={J. Combinat. Designs 11 (2003), 218-228},
    year={2002},
    archivePrefix={arXiv},
    eprint={math/0205303},
    primaryClass={math.CO cs.IT math.IT}
}