Linear and Circular Single Change Covering Designs Re-visited

TL;DR

This paper advances the theory of single change covering designs by establishing new existence results for circular designs with specific parameters using recursive and difference methods.

Contribution

It introduces a new recursive construction and proves the existence of circular single change covering designs for various parameters, filling gaps in the theoretical understanding.

Findings

Established existence of circular c(k-1)+1,k,c^2(2k-2)+c designs for all c  1, k  2.

Solved the existence of circular v,4,b for all v.

Proved existence of three residue classes of v,5,b modulo 16.

Abstract

A \textbf{single change covering design} is a $v$-set $X$ and an ordered list $\cL$ of $b$ blocks of size $k$ where every $t$-set must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. 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$(v,5,b)$ modulo 16. We prove the existence of circular \sccd$(2c(k-1)+1,k,c^2(2k-2)+c)$, for all $c\geq 1, k\geq2 $, using difference methods.