Folded Reed-Solomon (FRS) codes are a well-studied family of codes, known for achieving list decoding capacity. In this work, we give improved deterministic and randomized algorithms for list decoding FRS codes of rate \(R\) up to radius \(1 - R - \varepsilon\).

The classical rank aggregation problem seeks to combine a set X of n permutations into a single representative "consensus" permutation. In this paper, we investigate two fundamental rank aggregation tasks under the well-studied Ulam metric: computing a median permutation (which minimizes the sum of Ulam distances to X) and computing a center permutation (which minimizes the maximum Ulam distance to X) in two settings. - Continuous Setting: In the continuous setting, the median/center is allowed to be any permutation. It is known that computing a center in the Ulam metric is NP-hard and we add to this by showing that computing a median is NP-hard as well via a simple reduction from the Max-Cut problem. While this result may not be unexpected, it had remained elusive until now and confirms a speculation by Chakraborty, Das, and Krauthgamer [SODA '21]. - Discrete Setting: In the discrete setting, the median/center must be a permutation from the input set. We fully resolve the fine-grained complexity of the discrete median and discrete center problems under the Ulam metric, proving that the naive Õ(n² L)-time algorithm (where L is the length of the permutation) is conditionally optimal. This resolves an open problem raised by Abboud, Bateni, Cohen-Addad, Karthik C. S., and Seddighin [APPROX '23]. Our reductions are inspired by the known fine-grained lower bounds for similarity measures, but we face and overcome several new highly technical challenges.

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 ρ > 0, there exists some ϵ > 0, and an infinite set Γ ⊆ N, such that for alln∈ Γ, there is an explicit setCnof (n!)ϵ many permutations on [n], such that every pair of permutations inCnhas pairwise Ulam distance at least (1 − ρ) ·n. Moreover, we can compute theithpermutation inCnin poly(n) time and can also decode in poly(n) time, a permutation π on [n] to its closest permutation π∗ inCn, if the Ulam distance of π and π∗ is less than (1−ρ)·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 leastn/6 · (1 − ρ), for any ρ > 0, and the bound on the distance can be improved ton/4 · (1 − ρ) if the construction of Goldreich andWigderson is directly analyzed in the 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.