Let G(x) be the generator polynomial used for CRC checking. What is the…

Concept: For a polynomial P(x) over GF(2) (coefficients 0/1, arithmetic mod 2), evaluating P(1) gives the parity of its coefficient sum: P(1) = 0 if P(x) has an even number of nonzero terms, and P(1) = 1 if it has an odd number. In CRC, an error pattern goes undetected exactly when its error polynomial E(x) is an exact multiple of the generator polynomial G(x). This parity fact is the lever used to guarantee detection of an entire class of error patterns at once, rather than checking each pattern individually.

1. An error pattern with an odd number of flipped bits has an error polynomial E(x) with an odd number of nonzero terms (one term per flipped bit position), so E(1) = 1.

2. If G(x) is required to have (1+x) as a factor, then G(1) = 0. Any polynomial that is a multiple of G(x), say Q(x)·G(x), evaluates to Q(1)·G(1) = Q(1)·0 = 0 at x = 1.

3. Since E(1) = 1 (not 0) for every odd-weight error, no odd-weight E(x) can ever be an exact multiple of G(x). So the CRC remainder can never come out zero for such an error — it is always detected, for any odd count of bit errors.

Checking the other options:

• “G(x) contains more than two terms” guarantees only that a single isolated bit error is detected (G(x) is not a lone monomial); it is a different, weaker guarantee that does not address the parity property odd-count detection needs.

• “G(x) does not divide 1+xk for k up to the frame length” is the standard guarantee for catching double-bit errors — a different error class, unrelated to the parity argument above.

• “G(x) has an odd number of terms” is the opposite parity: it forces G(1) = 1, so G(x) is NOT divisible by (1+x), and some odd-weight error polynomials can then slip through as exact multiples of G(x) — odd-numbered errors are no longer guaranteed to be caught.

Answer: Therefore, the condition that guarantees detection of every odd number of bit errors is that (1+x) be a factor of the generator polynomial G(x).