Sure, I can help you create a test case generator for the given problem statement. Below is the Python code for the test case generator:

```python
import random

def tcgen():
    # Define constraints
    N = random.randint(2, 10**5)
    L = random.randint(1, 10**9)
    
    # Generate the lengths of the ropes
    a = [random.randint(1, 10**9) for _ in range(N)]
    
    return N, L, a

# Example usage
for _ in range(5):  # Generate 5 test cases
    N, L, a = tcgen()
    print(f"Input:")
    print(f"{N} {L}")
    print(" ".join(map(str, a)))
    print()
```

This code snippet generates random test cases for the problem. Here is a breakdown of the code:

1. **Define Constraints**: The number of ropes \( N \) is randomly chosen between 2 and \( 10^5 \). The minimum length \( L \) is randomly chosen between 1 and \( 10^9 \).

2. **Generate Lengths of the Ropes**: The lengths of the ropes are generated as a list \( a \) with \( N \) elements, each randomly chosen between 1 and \( 10^9 \).

3. **Return the Generated Test Case**: The function returns the values of \( N \), \( L \), and \( a \).

4. **Example Usage**: The example at the end demonstrates how to use the `tcgen` function to generate and print 5 test cases.

You can run this script to generate as many test cases as you need for the problem.