A single-change covering design (SCCD) is a $v$-set $X$ and an ordered list $\mathrm{ℒ}$ of $b$ blocks of size $k$ where every pair from $X$ must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given $v,k$. In this paper, we use a new recursive construction to solve the existence of circular SCCD($v,4,b$) for all $v$ and three residue classes of circular SCCD($v,5,b$) modulo 16. We solve the existence of three residue classes of SCCD$\left(v,5,b\right)$ modulo 16. We prove the existence of circular SCCD$\left(2c\left(k-1\right)+1,k,c^2\left(2k-2\right)+c\right)$, for all $c\ge 1,k\ge 2$, using difference methods.

中文翻译：

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重新审视线性和圆形单一变化覆盖设计

单次变化覆盖设计(SCCD) 是$v$-放$X$和一个有序列表$\mathrm{ℒ}$的$乙$大小块$k$每对来自哪里$X$必须至少出现在一个块中。每对连续块仅相差一个元素。这是线性单变化覆盖设计，或者更简单地说，单变化覆盖设计。当第一个和最后一个块也有一个元素不同时，单次更改覆盖设计是圆形的。如果对于给定的情况无法构造其他更小的设计，则单次更改覆盖设计是最小的$v,k$。在本文中，我们使用一种新的递归构造来解决循环SCCD($v,4,乙$） 对全部$v$以及圆形 SCCD 的三个残基类别（$v,5,乙$) 模 16。我们求解 SCCD 的三个残差类别的存在性$（v,5,乙）$模 16。我们证明圆形 SCCD 的存在性$（2C（k-1）+1,k,C^2（2k-2）+C）$， 对全部$C\ge 1,k\ge 2$，使用差分方法。