In a cyclic redundancy check (CRC), the receiver declares “no error” when the computed remainder over the received frame is equal to what value?

Introduction / Context:
CRC is one of the most widely used error-detection mechanisms at the data link layer. Sender and receiver agree on a generator polynomial; the sender appends check bits (the remainder) so that the transmitted codeword is divisible by the generator. The receiver rechecks divisibility to detect corruption.

Given Data / Assumptions:

• Standard CRC framing: data bits followed by r check bits.
• Receiver recomputes division by the same generator.
• Bit errors are modeled as additions modulo 2.

Concept / Approach:
The sender transmits a codeword C(x) such that C(x) mod G(x) = 0. If the channel introduces no error, the receiver’s division also yields remainder 0. A nonzero remainder indicates that the received polynomial is not a valid codeword under the agreed generator, so an error is flagged.

Step-by-Step Solution:
Sender computes remainder R so that (Data * x^r + R) / G has remainder 0.Receiver divides the entire received sequence by G.If the remainder equals 0, declare “no error” (within CRC’s detection capability).If the remainder is nonzero, declare an error and request retransmission.

Verification / Alternative check:
Try a small example with G = 1011 on a short data word; when check bits are appended correctly and no corruption occurs, the receiver’s modulo-2 division yields a remainder of 000.

Why Other Options Are Wrong:
the quotient at the sender: Quotient is not checked; remainder is.

nonzero: Nonzero remainder signals an error, not correctness.

equal to the remainder at the sender: The sender’s remainder is appended; the receiver expects an overall remainder of zero, not equality to a stored value.

None of the above: Incorrect because “zero” is correct.

Common Pitfalls:
Confusing the appended check field (sender’s remainder) with the receiver’s final remainder after dividing the entire codeword; the correct check is zero remainder at the receiver.

Final Answer:
zero