一个长度为n，半径为R的非对称二元覆盖码是n-立方体Q_n的子集C，使得Q_n中的每个向量x都可以由C中的某个向量c通过将c中至多R1变为0得到，其中R尽可能小。K^+（n，R）被定义为这种码的最小长度。利用非对称球覆盖界和概率方法证明了K^+（n，R）的阶为2^n/n^R，其中R为常数.我们证明了对常数Coradius R'，K^+（n，n-R'）= R '+1当且仅当n>= R'（R '+1）/2。这两个结果分别推广到近常数R和R '。K^+的各种界限是根据最小码中0或1的总数给出的。最小非对称线性二进制码（[n，R]^+码）的维数被确定为min（0，n-R）。最后，我们讨论了公开问题和计算K^+的显式值的技术，并给出了一个最佳已知边界表。
英文摘要：
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.
Further results on binary linear covering codes and saturating sets in projective spaces
Kharkevich Institute for Information Transmission Problems Russian Academy of Sciences（信息传输问题哈尔维奇研究所 俄罗斯科学院）；Department of Mathematics and Computer Science, University of Perugia（佩鲁贾大学数学与计算机科学系）；Independent（独立研究者）
Subspace coverings and generalized covering radii of generalized Zetterberg codes
Anhui University（安徽大学）；Nanyang Technological University（南洋理工大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Hefei University of Technology（合肥工业大学）
