Certainly! Below is the test case generator for the given problem statement:

```python
import random

def tcgen():
    # Generate values for N and L within the given constraints
    N = random.randint(2, 10**5)
    L = random.randint(1, 10**9)
    
    # Generate the list of rope lengths a such that each length is within the given constraints
    a = [random.randint(1, 10**9) for _ in range(N)]
    
    return N, L, a

# Example usage:
N, L, a = tcgen()
print(N)
print(L)
print(' '.join(map(str, a)))
```

To understand the structure of the test case generator:
- `N` is the number of ropes, which is randomly chosen between 2 and \(10^5\).
- `L` is the minimum length required to untie a knot, which is randomly chosen between 1 and \(10^9\).
- `a` is a list of rope lengths, where each length is randomly chosen between 1 and \(10^9\).

When using this generator, it will produce random valid inputs for the problem statement. Let's see an example of how the generated output might look:

```python
N, L, a = tcgen()
print(N)
print(L)
print(' '.join(map(str, a)))
```

Example output:
```
5
1000000000
123456789 987654321 456789123 789123456 123789456
```

This test case generator ensures that the generated inputs are within the problem constraints and can be used to test the solution to the problem effectively.