论文标题
双重变更覆盖设计的差异方法
Difference Methods for Double Change Covering Designs
论文作者
论文摘要
a \ textbf {double-change覆盖设计}（dccd）是一个$ v $ -set $ v $和订购的列表$ \ nathcal {l} $ $ b $ b $ size $ k $的$ b $块，其中$ v $中的每对必须在一个块中发生至少一个块，而每对连续块完全不同。如果它具有最少的块，而当第一个和最后一个块也不同于两个元素时，则是\ textbf {minimal}，而\ textbf {coundular}。我们提供了一种递归结构，该结构使用1次构造和扩展集来构建DCCD（$ V+\ frac {V+K-2} {K-2} {K-2}，K，B+\ frac {V} {k-2} \ frac {V+K-2} {V+K-2} {2K-4} $）我们从单个变更覆盖设计中构建了圆形DCCD（$ 2K-2，K，K-1 $）和圆形DCCD（$ 2K-3，K，K-2 $），并在$ v = 2k-2 $时确定最小的DCCD。我们使用差异方法来构建最小圆形DCCD（$ c（4k-6）+1，k，c^2（4K-6）+c $）的五个无限族，任何$ c \ leq 5 $对于任何$ k \ geq 3 $。然后，递归结构用于建造这些无限家庭成员的十二个最小DCCD。最后，差异方法用于构建最小的圆形DCCD（61,4,366）。
A \textbf{double-change covering design} (DCCD) is a $v$-set $V$ and an ordered list $\mathcal{L}$ of $b$ blocks of size $k$ where every pair from $V$ must occur in at least one block and each pair of consecutive blocks differs by exactly two elements. It is \textbf{minimal} if it has the fewest block possible and \textbf{circular} when the first and last blocks also differ by two elements. We give a recursive construction that uses 1-factorizations and expansion sets to construct a DCCD($v+\frac{v+k-2}{k-2},k,b+\frac{v}{k-2}\frac{v+k-2}{2k-4}$) from a DCCD($v,k,b$). We construct circular DCCD($2k-2,k,k-1$) and circular DCCD($2k-3,k,k-2$) from single change covering designs and determine minimal DCCD when $v=2k-2$. We use difference methods to construct five infinite families of minimal circular DCCD($c(4k-6)+1,k,c^2(4k-6)+c$) when $c\leq 5$ for any $k\geq 3$. The recursive construction is then used to build twelve additional minimal DCCD from members of these infinite families. Finally the difference method is used to construct a minimal circular DCCD(61,4,366).
