Sure! To generate test cases for the given problem statement, we need to take into account the constraints and the format of the input. Here is a test case generator for the problem of counting valid ways to split a string into an IPv4 address:

```python
import random

def generate_ip_string():
    length = random.randint(4, 12)  # |S| should be between 4 and 12
    S = ''.join(random.choices('0123456789', k=length))
    return S

def tcgen():
    S = generate_ip_string()
    return S

# Example usage
if __name__ == "__main__":
    for _ in range(5):  # Generate 5 test cases
        print(tcgen())
```

This generator function creates a string \( S \) of digits with a length between 4 and 12, which conforms to the constraints provided in the problem statement. This should help generate diverse and random test cases to test the solution's robustness.

Here's a brief explanation of the code:
1. **generate_ip_string()**: This function generates a random string of digits of a length between 4 and 12.
2. **tcgen()**: This function calls `generate_ip_string()` to get the test string \( S \) and returns it.
3. **Example usage**: This section demonstrates how to use the `tcgen()` function to generate multiple test cases. In this example, it generates 5 test cases and prints them.