Explicit Good Codes Approaching Distance 1 in Ulam Metric

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
$\alpha > 0$
, and an infinite set
$\Gamma\subseteq \mathbb{N}$
, such that for all
$n\in \Gamma$
, there is an explicit set
$C_{n}$
of
$(n!)^{\alpha}$
many permutations on
$[n]$
, such that every pair of permutations in
$C_{n}$
has pairwise Ulam distance at least
$(1-\epsilon)\cdot n$
. 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
$\pi$
on
$[n]$
to its closest permutation
$\pi^{*}$
in
$C_{n}$
, if the Ulam distance of
$\pi$
and
$\pi^{*}$
is less than
$\frac{(1-\varepsilon)n}{4}$
. 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
$\frac{n}{6}\cdot(1-\varepsilon)$
, 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.