Hızlı yanıt: kod örneği
I think this is the issue:An $(8, 5, 4)$ covering design is a set $\mathcal{B}$ of subsets of $\{1, 2, \ldots, 8\}$ (hereafter I'll use $[8]$ to refer to $\{1,2, \ldots, 8\}$) such that any $4$-element subset of $[8]$ is a subset of some $B \in \mathcal{B}$.What you have is not an $(8, 5, 4)$ design: for example, $\{1, 2, 4, 7\}$ isn't a subset of any of your five $5$-element subsets of $[8]$.But, your sets are enough to guarantee (and I had Sage help me make sure!) that for all $5$-element subsets of $[8]$, at least one of its $4$-element subsets is a subset of one of your sets.This is not my area of expertise, so I'm not aware of any good references for the sort of things you're looking to construct. But, they are not $(v, k, t)$ covering designs.From the answer to your other question:A Covering Design $C(v,k,t,m,l,b)$ is a pair $(V,B)$, where $V$is a set of $v$ elements (called points) and $B$ is a collection of $b$ $k$-subsets of $V$ (called blocks), such that every $m$-subset of $V$ intersects at least $l$ members of $B$ in at least $t$ points. It is required that $v \geq k \geq t$ and $m \geq t.$You want $l=1,k=5,$ $V=\{1,2,\ldots,14\} \quad(v=14)$ etc. Google the "La Jolla Covering Repository" for extensive tables.We can reconcile this with the definition given by La Jolla:A $(v,k,t)-$covering design is a collection of $k$-element subsets, called blocks, of $\{1,2,\ldots ,v\}$, such that any $t$-element subset is contained in at least one block.If we are talking about a $(v, k, t)$ covering design, it fits into the above definition, but it's suppressed some parameters: Namely, requiring every $t$-element subset to be contained in at least one block, we're stipulating that $m = t$ (in your case, $4$).Specifically, your sets are a
$$C(v = 8,\, k = 5,\, t = 4,\, m = 5,\, l = 1,\, b = 5)$$
covering design. But what's listed in La Jolla is a
$$C(v = 8,\, k = 5,\, t=4,\, m=4,\, l=1,\, b=20)$$covering design. I suspect that you need $b \ge 20$ subsets to cover $m = t = 4$-element subsets of $[8]$ (hence the $20$ sets in the repository) while, evidently, covering $m = 4$-element subsets of all $t = 5$-element subets of $[8]$ requires fewer sets; at most $b = 5$, as your collection shows.