The exception to this result is a bit pattern the same as that of the generator polynomial.

All uneven bit errors are detected by generator polynomials with even number of terms.

The degree of the generator polynomial is 1, so we first multiplied the message by to get .

Moreover, if and , the generator polynomial has degree at most .

If the generator polynomial g has degree d then the rank of the code C is .

Suppose that the generator polynomial has degree .

Analysis Technique using bitfilters allows one to very efficiently determine the properties of a given generator polynomial.

Here is the original message polynomial and is the degree- generator polynomial.

An even code should have a generator polynomial that include (1+x) minimal polynomial as a product.

The ideal is generated by the unique monic element in C of minimum degree, the generator polynomial g.

All single bit errors within the bitfilter period mentioned above (for even terms in the generator polynomial) can be identified uniquely by their residual.

If the generator polynomial is primitive, then the resulting code has Hamming distance at least 3, provided that .

To define a cyclic code, we pick a fixed polynomial, called generator polynomial.

The generator polynomial of a BCH code has degree at most .

An -burst-error correcting Fire Code is defined by the following generator polynomial: .

The length of the remainder is always less than the length of the generator polynomial, which therefore determines how long the result can be.

Consider the polynomial code over with , , and generator polynomial .

Every cyclic code with generator polynomial of degree can detect all bursts of length .

Since the generator polynomial is of degree 4, this code has 11 data bits and 4 checksum bits.

Fix integers and let be some fixed polynomial of degree , called the generator polynomial.

The checksum is a 32 bit value calculated according to the generator polynomial represented by 0x104C11DB7:

All the well-known CRC generator polynomials of degree have two common hexadecimal representations.

The selection of the generator polynomial is the most important part of implementing the CRC algorithm.

A polynomial code is cyclic if and only if the generator polynomial divides .

The maximum length is , when is the degree of the generator polynomial (which itself has a length of ).