Explicit Good Codes Approaching Distance 1 in Ulam Metric

The Ulam distance between two permutations on $[n]$ is n minus the length of their longest common subsequence. In this paper, we show that for every $\rho \gt 0$ , there exists some $\epsilon \gt 0$ , and an infinite set $\Gamma \subseteq \mathbb {N}$ , such that for all $n\in \Gamma $ , there is an explicit set $C_{n}$ of $(n!)^{\epsilon }$ many permutations on $[n]$ , such that every pair of permutations in $C_{n}$ has pairwise Ulam distance at least $(1-\rho)\cdot n$ . Moreover, we can compute the $i^{\text {th}}$ permutation in $C_{n}$ in $\mathrm {poly}(n)$ time and can also decode in $\mathrm {poly}(n)$ time, a permutation $\pi $ on $[n]$ to its closest permutation $\pi ^{*}$ in $C_{n}$ , if the Ulam distance of $\pi $ and $\pi ^{*}$ is less than $ \frac {(1-\rho)\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 $(n!)^{\Omega (1)}$ many permutations on $[n]$ , such that every pair of them have pairwise Ulam distance at least $\frac {n}{6}\cdot (1-\rho)$ , for any $\rho \gt 0$ , and the bound on the distance can be improved to $\frac {n}{4}\cdot (1-\rho)$ if the construction of Goldreich and Wigderson is directly analyzed in the Ulam metric.