  The Transposition Distance Problem (TDP) is a classical problem in genome
rearrangements which seeks to determine the minimum number of transpositions
needed to transform a linear chromosome into another represented by the
permutations $\pi$ and $\sigma$, respectively. This paper focuses on the
equivalent problem of Sorting By Transpositions (SBT), where $\sigma$ is the
identity permutation $\iota$. Specifically, we investigate palisades, a family
of permutations that are "hard" to sort, as they require numerous
transpositions above the celebrated lower bound devised by Bafna and Pevzner.
By determining the transposition distance of palisades, we were able to provide
the exact transposition diameter for $3$-permutations (TD3), a special subset
of the Symmetric Group $S_n$, essential for the study of approximate solutions
for SBT using the simplification technique. The exact value for TD3 has
remained unknown since Elias and Hartman showed an upper bound for it. Another
consequence of determining the transposition distance of palisades is that,
using as lower bound the one by Bafna and Pevzner, it is impossible to
guarantee approximation ratios lower than $1.375$ when approximating SBT. This
finding has significant implications for the study of SBT, as this problem has
been subject of intense research efforts for the past 25 years.
