Discussion
Which of the following CRC generator guarantee the detection of a single bit errors ?
Answer: D
The generator that guarantees detecting a single-bit error is \(X^{2} + 1\).
A single-bit error at position i is the error polynomial \(x^{i}\). It goes undetected only if the generator divides \(x^{i}\). A generator with at least two terms, one of them the constant 1, cannot divide any \(x^{i}\), so it catches every single-bit error.
\(X^{2} + 1\) has two terms, one of them 1, so it qualifies. The generator 1 divides every polynomial, so every remainder is zero and no error at all is detected. \(X^{4} + X^{2} = X^{2}(X^{2} + 1)\) and \(X^{3} + X = X(X^{2} + 1)\) have no constant term, so they fail the condition and the guarantee does not hold for them.
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