Problem Analysis #8

[THOUGHT] To determine the maximum size of a ball of radius 1 in $\mathbb{Z}_2^{100}$ using the Fixed Length Levenshtein distance, we need to identify all vectors $\mathbf{v}$ that can be transformed from a center vector $\mathbf{x}$ through exactly one deletion and one insertion.

Given a vector $\mathbf{x} = (x_1, x_2, \ldots, x_{100}) \in \mathbb{Z}_2^{100}$, we need to count the number of ways to:

1. Delete one element from $\mathbf{x}$, resulting in a vector of length 99.
2. Insert one element into the resulting vector of length 99 to get back to a vector of length 100.

Let's break this down:

1. Deletion: We can delete any one of the 100 elements of $\mathbf{x}$. This gives us 100 choices.
2. Insertion: After deleting an element, we have a vector of length 99. We can insert either a 0 or a 1 at any one of the 100 positions (including the beginning and the end). This gives us 100 positions and 2 choices for the value to insert.

Thus, for each of the 100 deletions, we have 200 possible insertions (100 positions * 2 values). However, we must be careful not to double-count the original vector $\mathbf{x}$, as it should not be included in the ball of radius 1 around itself.

Therefore, the total number of vectors $\mathbf{v}$ that can be transformed from $\mathbf{x}$ through one deletion and one insertion is: $100 \times 200 = 20000$

However, this count includes the original vector $\mathbf{x}$ multiple times, since inserting the same value back into the same position after deletion results in $\mathbf{x}$. Each deletion-insertion pair that results in $\mathbf{x}$ corresponds to:

• Deleting $x_i$ and inserting $x_i$ back at the $i$-th position (100 such cases).

Thus, we need to subtract these 100 cases from our total count: $20000 - 100 = 19900$

So, the maximum size of the ball of radius 1 in $\mathbb{Z}_2^{100}$ centered at $\mathbf{x}$ is: $\boxed{19900}$

This calculation does not require a Python code execution because it is a straightforward combinatorial problem.

[/THOUGHT] \boxed{19900}