Final answer:
Yes, an error in bit 256 can be detected using CRC-16. CRC uses a polynomial to generate a checksum that's appended to the data, which is then checked at the receiver's end to detect errors; a non-zero remainder in the division indicates an error.
Step-by-step explanation:
The student has asked whether a transmission error in bit 256 can be detected using CRC-16 with the polynomial x¹⁶ + x¹⁵ + x² + 1. The answer is yes, the error can be detected. CRC, or Cyclic Redundancy Check, is an error-detecting code commonly used in networks and storage devices to detect accidental changes to raw data. In CRC, a sequence of redundant bits, known as the CRC or the CRC checksum, is appended to the end of a data unit so that the resulting data unit becomes exactly divisible by a predetermined binary number.
For a data unit of 256 bytes (2048 bits), the CRC-16 will append 16 extra bits to make the transmitted frame. If bit 256 has an error, meaning that it has changed from 0 to 1 or vice versa, the calculation on the receiver's side will give a non-zero remainder when it divides the modified message by the CRC-16 polynomial. This non-zero remainder signifies that an error has been detected.
Mathematically, the received message M(x) will have x⁰ (the least significant bit) changed if the error is in bit 256. The polynomial division by P(x) = x¹⁶ + x¹⁵ + x² + 1 used for CRC calculation will not divide M(x) + x⁰ without a remainder. Therefore, the error is detected.