A \((2,k,v)\) covering design is a pair \((X,{\mathcal F})\) such that \(X\) is a \(v\)-element set and \({\mathcal F}\) is a family of \(k\)-element subsets, called blocks, of \(X\) with the property that every pair of distinct elements of \(X\) is contained in at least one block. Let \(C(2,k,v)\) denote the minimum number of blocks in a \((2,k,v)\) covering design. We construct in this paper a class of \((2,k,v)\) covering designs using number theoretic means, and determine completely the functions \(C(2,6,6^ n\cdot 28)\) for all \(n\geq 0\), and \(C(2,6,6^ n \cdot 28-5)\) for all \(n \geq 1\). Our covering designs have interesting combinatorial properties.