The combination of these factors means that good CRC polynomials are often primitive polynomials ( which have the best 2-bit error detection ) or primitive polynomials of degree n-1, multiplied by x + 1 ( which detects all odd numbers of bit errors, and has half the two-bit error detection ability of a primitive polynomial of degree n ).
32.
The combination of these factors means that good CRC polynomials are often primitive polynomials ( which have the best 2-bit error detection ) or primitive polynomials of degree n-1, multiplied by x + 1 ( which detects all odd numbers of bit errors, and has half the two-bit error detection ability of a primitive polynomial of degree n ).
33.
From Gauss'lemma it follows that is reducible in as well, and in fact can be written as the product " GH " } } of two non-constant polynomials ( in case is not primitive, one applies the lemma to the primitive polynomial ( where the integer is the content of ) to obtain a decomposition for it, and multiplies into one of the factors to obtain a decomposition for ).
34.
The cyclic redundancy check ( CRC ) is an error-detection code that operates by interpreting the message bitstring as the coefficients of a polynomial over GF ( 2 ) and dividing it by a fixed generator polynomial also over GF ( 2 ); see Mathematics of CRC . Primitive polynomials, or multiples of them, are sometimes a good choice for generator polynomials because they can reliably detect two bit errors that occur far apart in the message bitstring, up to a distance of for a degree " n " primitive polynomial.
35.
The cyclic redundancy check ( CRC ) is an error-detection code that operates by interpreting the message bitstring as the coefficients of a polynomial over GF ( 2 ) and dividing it by a fixed generator polynomial also over GF ( 2 ); see Mathematics of CRC . Primitive polynomials, or multiples of them, are sometimes a good choice for generator polynomials because they can reliably detect two bit errors that occur far apart in the message bitstring, up to a distance of for a degree " n " primitive polynomial.
36.
Note first that in " F " [ " X " ] \ { 0 } any class of associate elements ( whose elements are related by multiplication by nonzero elements of the field " F " ) meets the set of primitive elements in " R " [ " X " ] : starting from an arbitrary element of the class, one can first ( if necessary ) multiply by a nonzero element of " R " to enter into the subset " R " [ " X " ] ( removing denominators ), then divide by the greatest common divisor of all coefficients to obtain a primitive polynomial.
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