Linear and circular single-change covering designs revisited

Abstract

A single-change covering design (SCCD) is a $v$-set $X$ and an ordered list $ℒ$ 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.