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

```python
import random

def tcgen():
    N = random.randint(2, 10**5)
    L = random.randint(1, 10**9)
    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)))
```

This code will generate a random test case for the problem by following these steps:
1. Randomly choose the number of ropes \( N \) between 2 and \( 10^5 \).
2. Randomly choose the minimum length \( L \) between 1 and \( 10^9 \).
3. Generate a list \( A \) of \( N \) integers, each between 1 and \( 10^9 \), representing the lengths of the ropes.

You can run the `tcgen` function to get randomized inputs for testing purposes in competitive programming.