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| Natura: | Preprint |
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2022
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| Accesso online: | https://arxiv.org/abs/2209.11010 |
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| _version_ | 1866913574568329216 |
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| author | Chafee, Amanda Stevens, Brett |
| author_facet | Chafee, Amanda Stevens, Brett |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_11010 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Linear and Circular Single Change Covering Designs Re-visited Chafee, Amanda Stevens, Brett Combinatorics 05B40, 05B10 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. |
| title | Linear and Circular Single Change Covering Designs Re-visited |
| topic | Combinatorics 05B40, 05B10 |
| url | https://arxiv.org/abs/2209.11010 |