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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.22607 |
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| _version_ | 1866912486659194880 |
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| author | Burgess, Andrea C. Danziger, Peter Horsley, Daniel Javed, Muhammad Tariq |
| author_facet | Burgess, Andrea C. Danziger, Peter Horsley, Daniel Javed, Muhammad Tariq |
| contents | Given positive integers $v$, $k$, $t$ and $λ$ with $v \geq k \geq t$, a packing design PD$_λ(v,k,t)$ is a pair $(V,\mathcal{B})$, where $V$ is a $v$-set and $\mathcal{B}$ is a collection of $k$-subsets of $V$ such that each $t$-subset of $V$ appears in at most $λ$ elements of $\mathcal{B}$. When $λ=1$, a PD$_1(v,k,t)$ is equivalent to a binary code with length $v$, minimum distance $2(k-t+1)$ and constant weight $k$. The maximum size of a PD$_λ(v,k,t)$ is called the {packing number}, denoted PDN$_λ(v,k,t)$. In this paper we consider packing designs with $k$ large relative to $v$. We prove that for a positive integer $n$, PDN$_λ(v,k,t) = n$ whenever $nk-(t-1)\binom{n}{λ+1} \leq λv < (n+1)k-(t-1)\binom{n+1}{λ+1}$.
We also prove that if no point appears in more than three blocks, then the blocks of a PD$_2(v,k,2)$ can be ordered so that no ordered pair occurs more than once. This produces a directed packing design and we show that the corresponding directed packing number is equal to $n$ when $nk-\binom{n}{3} \leq 2v < (n+1)k-\binom{n+1}{3}$. Such directed packing designs yield $(k-t)$-insertion/deletion codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_22607 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Packing Designs with large block size Burgess, Andrea C. Danziger, Peter Horsley, Daniel Javed, Muhammad Tariq Combinatorics 05B40 Given positive integers $v$, $k$, $t$ and $λ$ with $v \geq k \geq t$, a packing design PD$_λ(v,k,t)$ is a pair $(V,\mathcal{B})$, where $V$ is a $v$-set and $\mathcal{B}$ is a collection of $k$-subsets of $V$ such that each $t$-subset of $V$ appears in at most $λ$ elements of $\mathcal{B}$. When $λ=1$, a PD$_1(v,k,t)$ is equivalent to a binary code with length $v$, minimum distance $2(k-t+1)$ and constant weight $k$. The maximum size of a PD$_λ(v,k,t)$ is called the {packing number}, denoted PDN$_λ(v,k,t)$. In this paper we consider packing designs with $k$ large relative to $v$. We prove that for a positive integer $n$, PDN$_λ(v,k,t) = n$ whenever $nk-(t-1)\binom{n}{λ+1} \leq λv < (n+1)k-(t-1)\binom{n+1}{λ+1}$. We also prove that if no point appears in more than three blocks, then the blocks of a PD$_2(v,k,2)$ can be ordered so that no ordered pair occurs more than once. This produces a directed packing design and we show that the corresponding directed packing number is equal to $n$ when $nk-\binom{n}{3} \leq 2v < (n+1)k-\binom{n+1}{3}$. Such directed packing designs yield $(k-t)$-insertion/deletion codes. |
| title | Packing Designs with large block size |
| topic | Combinatorics 05B40 |
| url | https://arxiv.org/abs/2410.22607 |