Final answer:
To calculate the CRC using polynomial long division, extend the message by appending zeros equal to the number of bits in the generator polynomial minus one, then use binary division to find the remainder. This remainder is the CRC.
Step-by-step explanation:
To calculate the Cyclic Redundancy Check (CRC) for the packet with the bit sequence 101010111101 using the generator polynomial C(x) = x⁵ + x² + x + 1, you first need to apply polynomial long division. The process resembles long division with numbers, but using binary operations. The bit sequence, or message, must initially be extended by the number of bits in the generator polynomial minus one. In this case, the generator polynomial has 6 bits (from x⁵ to x⁵⁰), so we append five zeros to the message.
The extended message is now 10101011110100000. Using binary division, we divide this by our generator polynomial, represented in binary as 100101. We perform the division by subtracting (via XOR operation) whenever the leading digit of our polynomial can divide the current portion of our message. The remainder of this division, once all possible subtractions have been made, is the CRC.
For example, we align the highest term of the polynomial with the leftmost bit of the message and perform XOR. The extended message changes as we move down the sequence, continuing the process until we reach the end. The resulting remainder is the desired CRC.
It's important to note that for practical applications, there may be more efficient computational methods, such as using lookup tables or hardware-implemented algorithms, which wouldn't require manual binary division. However, understanding the division process is critical to grasping the principles behind CRC.