Polynomial Codes, continued

What types of errors can (and cannot) be detected with CRCs?

Let the error bits in T_arrived(x) be E(x). Each bit in E(x) corresponds to a bit that has been inverted.
If there are k 1 bits in E(x), then there have been k single-bit errors.

Then, due to the properties of the Modulo-2 arithmetic:

    (T(x) + E(x)) / G(x) = (T(x) / G(x)) + (E(x) / G(x))
                         =       0       + (E(x) / G(x))

We notice that if G(x) divides E(x) (because G(x) is a factor of E(x)), then errors can go by undetected.

Some important results:

• If G(x) has two or more terms all single bit errors can be detected.
• To detect double errors, G(x) must not be divisible by ( x ^k + 1) for k up to some maximum frame length.
• To detect an odd number of errors, G(x) should contain (x+1) as a factor.
• Polynomials of degree r will detect all burst errors of <= r bits.