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) (2n /nR ) 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.

• Type: Journal
• Year: 2002
• Where: JCT
• Authors: Joshua N. Cooper, Robert B. Ellis, Andrew B. Kahng
• Keywords: Asymmetric Sphere-covering Bound, Code, JCT 2002, N-cube Qn