Asymmetric Binary Covering Codes

An asymmetric binary covering code of length n and radius R is a subset C of the n-cube Qn such that every vector x ∈ Qn can be obtained from some vector c ∈ C by changing at most R 1’s of c to 0’s, where R is as small as possible. K(n,R) is defined as the smallest size of such a code. We show K(n,R) ∈ Θ(2/n) for constant R, using an asymmetric sphere-covering bound and probabilistic methods. We show K(n, n−R) = R+1 for constant coradius R iff n ≥ R(R+1)/2. These two results are extended to near-constant R and R, respectively. Various bounds on K are given in terms of the total number of 0’s or 1’s in a minimal code. The dimension of a minimal asymmetric linear binary code ([n,R]-code) is determined to be min{0, n−R}. We conclude by discussing open problems and techniques to compute explicit values for K, giving a table of best known bounds.