In this code, we will model the end-to-end error detection process using CRC: Message → CRC Encoding → Transmission (with noise) → CRC Decoding → Error Detection.

Problem statement

In digital communication systems, error detection is critical to ensure data integrity when messages are transmitted over unreliable channels. One of the most widely used techniques is the Cyclic Redundancy Check (CRC), which appends a set of check bits (CRC bits) to the original message so that the receiver can detect whether errors occurred during transmission.

This problem involves implementing and simulating CRC-based error detection in C++. The program should:

1. Accept a binary message from the user (e.g., "1101011011") representing the data bits to be transmitted. Use a function to convert the string to vector<bool> internally. Another function should convert the vector<bool> back to a binary string for display purposes.
2. Accept a generator polynomial, i.e., the divisor, (also in binary form, e.g., "10011") used for CRC computation.
   • The generator polynomial must be non-empty and start with 1.
3. Compute the CRC bits by performing polynomial division of the message (padded with zeros) by the generator polynomial. The remainder of this division forms the CRC.
4. Append the CRC bits to the message to create the transmitted codeword.
5. Simulate transmission over a noisy channel by flipping each transmitted bit with a user-defined probability (between 0.0 and 1.0). This models random corruption in the channel.
6. Display both transmitted and received bits to allow comparison.
7. Perform CRC error checking at the receiver side by dividing the received bits with the generator polynomial.
   • If the remainder is all zeros, the data is considered error-free.
   • If the remainder contains any nonzero bits, an error has been detected.

Inputs:

• Message bits (binary string, e.g., "1101011011").
• Generator polynomial (binary string, e.g., "10011").
• Bit corruption probability (floating-point number between 0.0 and 1.0).

Outputs:

• Message length and generator polynomial length.
• Computed CRC bits.
• Transmitted codeword (message + CRC).
• Transmitted bits after corruption and received bits.
• CRC remainder at receiver side.
• Status of error detection (either “CRC check passed” or “CRC check failed”).