Linear and circular single-change covering designs revisited

A single-change covering design (SCCD) is a v $v$-set X $X$ and an ordered list Script capital L ${\rm{ {\mathcal L} }}$ of b $b$ blocks of size k $k$ where every pair from X $X$ 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 $v,k$. In this paper, we use a new recursive construction to solve the existence of circular SCCD(v,4,b $v,4,b$) for all v $v$ and three residue classes of circular SCCD(v,5,b $v,5,b$) modulo 16. We solve the existence of three residue classes of SCCD(v,5,b) $(v,5,b)$ modulo 16. We prove the existence of circular SCCD(2c(k-1)+1,k,c2(2k-2)+c) $(2c(k-1)+1,k,{c}<^>{2}(2k-2)+c)$, for all c >= 1,k >= 2 $c\ge 1,k\ge 2$, using difference methods.