Problem Analysis #8

[THOUGHT] We need the size (cardinality) of the “ball” of radius 1 under the Fixed Length Levenshtein distance in $\mathbb{Z}_2^{100}$, i.e., all vectors $y\in\mathbb{Z}_2^{100}$ such that the distance between $y$ and the center $x$ is $\le 1$. Here a deletion removes one coordinate from the first vector, and an insertion adds one coordinate to the first vector; to keep the length fixed, one deletion must be paired with one insertion. Thus a distance-1 transformation consists of choosing one coordinate to delete from $y$ and one coordinate (possibly the same) to insert, yielding $x$. In $\mathbb{Z}_2$, each coordinate is either 0 or 1. For any $y$, there is exactly one deletion that makes the resulting length-99 vector match the length-99 vector obtained from $x$ by deleting some coordinate, and then exactly one insertion to restore length 100. This forces $y$ to equal $x$. Therefore the radius-1 ball contains only the center itself, so its maximum size is 1. [/THOUGHT] [PYTHON] print(1) [/PYTHON]

[THOUGHT] The Python output is $1$, so the maximum size of the radius‑1 ball in $\mathbb{Z}_2^{100}$ is $1$. \boxed{1}