Asymmetric binary covering codes

An asymmetric binary covering code of length n and radius R is a subset C of the n-cube Q_n such that every vector x in Q_n can be obtained from some vector c in 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) is of order 2^n/n^R 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.Comment: 16 page

Simplifications to A New Approach to the Covering Radius...”

We simplify the proofs of four results in [3], restating two of them for greater clarity. The main purpose of this note is to give a brief transparent proof of Theorem 7 of [3], the main upper bound of that paper. The secondary purpose is to give a more direct statement and proof of the integer programming determination of covering radius of [3]. Theorem 7 of [3] follows from a simple result in [2], which we state with the notation (for the linear code A)