Sure! Here's a test case generator for the given problem statement:

```python
import random

def tcgen():
    N = random.randint(1, 10)
    bit_line_length = 2 ** N
    bit_line = ''.join(random.choice('01') for _ in range(bit_line_length))
    return N, bit_line

def main():
    N, bit_line = tcgen()
    print(N)
    print(bit_line)

if __name__ == "__main__":
    main()
```

Explanation:
- `N` is randomly chosen between 1 and 10, inclusive, as per the problem constraints.
- The length of `bit_line` is calculated as \(2^N\).
- `bit_line` is generated as a string of random '0' and '1' characters of the specified length.
- The generated test case is printed in the required format.

You can run the `main` function to generate and print a random test case. Here is how it works:
- It generates a random value for `N`.
- Computes the length of `bit_line` as \(2^N\).
- Constructs the `bit_line` by randomly choosing '0' or '1' for each character position.
- Prints out the values of `N` and `bit_line`.

This generator will help you create numerous test cases to effectively test your implementation of the logic for the provided problem statement.