Explicit Good Codes Approaching Distance 1 in Ulam Metric
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866911873739259904 |
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| author | Goldenberg, Elazar Habib, Mursalin S, Karthik C. |
| author_facet | Goldenberg, Elazar Habib, Mursalin S, Karthik C. |
| contents | The Ulam distance of two permutations on $[n]$ is $n$ minus the length of their longest common subsequence. In this paper, we show that for every $\varepsilon>0$, there exists some $α>0$, and an infinite set $Γ\subseteq \mathbb{N}$, such that for all $n\inΓ$, there is an explicit set $C_n$ of $(n!)^α$ many permutations on $[n]$, such that every pair of permutations in $C_n$ has pairwise Ulam distance at least $(1-\varepsilon)\cdot n$. Moreover, we can compute the $i^{\text{th}}$ permutation in $C_n$ in poly$(n)$ time and can also decode in poly$(n)$ time, a permutation $π$ on $[n]$ to its closest permutation $π^*$ in $C_n$, if the Ulam distance of $π$ and $π^*$ is less than $ \frac{(1-\varepsilon)\cdot n}{4} $.
Previously, it was implicitly known by combining works of Goldreich and Wigderson [Israel Journal of Mathematics'23] and Farnoud, Skachek, and Milenkovic [IEEE Transactions on Information Theory'13] in a black-box manner, that it is possible to explicitly construct $(n!)^{Ω(1)}$ many permutations on $[n]$, such that every pair of them have pairwise Ulam distance at least $\frac{n}{6}\cdot (1-\varepsilon)$, for any $\varepsilon>0$, and the bound on the distance can be improved to $\frac{n}{4}\cdot (1-\varepsilon)$ if the construction of Goldreich and Wigderson is directly analyzed in the Ulam metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17235 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit Good Codes Approaching Distance 1 in Ulam Metric Goldenberg, Elazar Habib, Mursalin S, Karthik C. Information Theory Computational Complexity Data Structures and Algorithms Combinatorics The Ulam distance of two permutations on $[n]$ is $n$ minus the length of their longest common subsequence. In this paper, we show that for every $\varepsilon>0$, there exists some $α>0$, and an infinite set $Γ\subseteq \mathbb{N}$, such that for all $n\inΓ$, there is an explicit set $C_n$ of $(n!)^α$ many permutations on $[n]$, such that every pair of permutations in $C_n$ has pairwise Ulam distance at least $(1-\varepsilon)\cdot n$. Moreover, we can compute the $i^{\text{th}}$ permutation in $C_n$ in poly$(n)$ time and can also decode in poly$(n)$ time, a permutation $π$ on $[n]$ to its closest permutation $π^*$ in $C_n$, if the Ulam distance of $π$ and $π^*$ is less than $ \frac{(1-\varepsilon)\cdot n}{4} $. Previously, it was implicitly known by combining works of Goldreich and Wigderson [Israel Journal of Mathematics'23] and Farnoud, Skachek, and Milenkovic [IEEE Transactions on Information Theory'13] in a black-box manner, that it is possible to explicitly construct $(n!)^{Ω(1)}$ many permutations on $[n]$, such that every pair of them have pairwise Ulam distance at least $\frac{n}{6}\cdot (1-\varepsilon)$, for any $\varepsilon>0$, and the bound on the distance can be improved to $\frac{n}{4}\cdot (1-\varepsilon)$ if the construction of Goldreich and Wigderson is directly analyzed in the Ulam metric. |
| title | Explicit Good Codes Approaching Distance 1 in Ulam Metric |
| topic | Information Theory Computational Complexity Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2401.17235 |