Asymmetric Binary Covering Codes

Abstract

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 2 Q n can be obtained from some vector c 2 C by changing at most R 1's of c to 0's, where R is as small as possible. K þ ðn; RÞ is defined as the smallest size of such a code. We show K þ ðn; RÞ 2 Yð2 n =n R Þ for constant R; using an asymmetric sphere-covering bound and probabilistic methods. We showThese two results are extended to near-constant R and % R R; respectively. Various bounds on K þ 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 (½n; R þ -code) is determined to be minf0; n À Rg:

On asymmetric coverings and covering numbers

“…Prompted by applications to the manufacture of semiconductor wafers, Cooper, Ellis and Kahng [7] have investigated the asymptotic behavior of D(n, R) for fixed R as n → ∞. In Section 2 of the present paper we show that the values of D(n, 1) for n ≤ 8 are as shown in Table 1.…”

“…Table 5 gives values of N (n, k, k −1, M ) together with a brief indication of how they were found. An entry such as 11,8,7 : 63(40); 64(1193)(based on [10,7,6, 46])…”

Computing marginals using MapReduce

“…However, we are still left with answering the question: how do we pack all the k th -order marginals into as few marginals of order m as possible, for each m > k. That will lead us to the matter of "asymmetric covering codes" or "covering numbers" [7,9].…”

“…Let n = 3, and let the three groups be A 1 A 2 A 3 , B 1 B 2 B 3 , and C 1 C 2 C 3 . From the first rule, we get the handles 9 2 /3 = 12.…”

О длине функций $k$-значной логики в классе полиномиальных нормальных форм по модулю $k$

“…Теперь оценим мощность покрытия множества E n k с помощью конструкции, обобща-ющей метод доказательства [10] верхней оценки мощности когда, асимметрически покрывающего булев куб.…”