Explicit Good Codes Approaching Distance 1 in Ulam Metric

Abstract

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.

Figures (2)

• Figure 1: In the figure, we consider setting where $N=3$. At stage 0, we have the identity permutation $(0,1,2,3,4,5,6,7)$. In stage 1, we decide to swap balls on the edge $000$ and $100$ and on the edge $011$ and $111$ (while not swapping the balls on the edge $010$ and $110$ and the edge $001$ and $101$). Thus, we obtain the permutation $(4,1,2,7,0,5,6,3)$. In stage 2, we decide to swap balls on the edge $000$ and $010$, the edge $001$ and $011$, and on the edge $100$ and $110$, obtaining the permutation $(2,7,4,1,6,5,0,3)$. Finally, in stage 2, we decide to swap balls only on the edge $110$ and $111$, obtaining the permutation $(2,7,4,1,6,5,3,0)$ which is our output. Note that all decisions were obtained as outcomes of some coin tosses.
• Figure 2: We illustrate in this figure the construction of a codeword described in Section \ref{['sec:construction']}, for the setting where $q=3, \ell=2, n=9, p=4, D = \{\sigma_0=012,\sigma_1=210, \sigma_2=102, \sigma_3=120 \},$ and $\textbf{w}=(w^{(1)}=301, w^{(2)}=223)$.

Theorems & Definitions (4)

• Theorem 1.1
• Claim 4.1
• proof
• proof