Covering, Packing, and List Codes

In this chapter we investigate two basic notions from Coding Theory: covering and packing. Usually if one considers a finite metric space, the main problem in Coding Theory (see, e.g., [L98]) is to find the maximal number of points in this space such that the balls of a given radius with centers in those points do not intersect. This is the packing problem. The dual problem in Coding Theory is the covering problem: find the minimal cardinality of a subset of the metric space such that the union of the balls with centers in the points of that set is the whole space. Usually the covering problem is much simpler than the packing problem (we see this in Lecture 8) and only asymptotic bounds on the cardinality of a packing are known in the general case.

The situation is quite different if one considers the packing and covering problems for k-uniform hypergraphs, when k is small (fixed). We see in the next lecture that in that case it is possible to find the exact asymptotical growth of the cardinalities of optimal coverings and packings (and they coincide).