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The CRC and associated polynomial typically have a name of the form CRC-n-XXX as in the table below. The simplest error-detection system, the parity bit, is in fact a 1-bit CRC: it uses the generator polynomial x + 1 (two terms), [5] and has the name CRC-1.
In the above equations, + + represents the original message bits 111, + is the generator polynomial, and the remainder (equivalently, ) is the CRC. The degree of the generator polynomial is 1, so we first multiplied the message by to get + +.
To maximise computation speed, an intermediate remainder can be calculated by first computing the CRC of the message modulo a sparse polynomial which is a multiple of the CRC polynomial. For CRC-32, the polynomial x 123 + x 111 + x 92 + x 84 + x 64 + x 46 + x 23 + 1 has the property that its terms (feedback taps) are at least 8 positions apart ...
Since the generator polynomial is of degree 10, this code has 5 data bits and 10 checksum bits. It is also denoted as: (15, 5) BCH code. (This particular generator polynomial has a real-world application, in the "format information" of the QR code.) The BCH code with = and higher has the generator polynomial
Here, codeword polynomial is an element of a linear code whose code words are polynomials that are divisible by a polynomial of shorter length called the generator polynomial. Every codeword polynomial can be expressed in the form c ( x ) = a ( x ) g ( x ) {\displaystyle c(x)=a(x)g(x)} , where g ( x ) {\displaystyle g(x)} is the generator ...
It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a generator polynomial, which is used as the divisor in a polynomial long division over a finite field, taking the input data as the dividend. The remainder becomes the result. A CRC has properties that make it well suited for detecting burst ...
By 1963 (or possibly earlier), J. J. Stone (and others) recognized that Reed–Solomon codes could use the BCH scheme of using a fixed generator polynomial, making such codes a special class of BCH codes, [4] but Reed–Solomon codes based on the original encoding scheme are not a class of BCH codes, and depending on the set of evaluation ...
Now, we can think of words as polynomials over , where the individual symbols of a word correspond to the different coefficients of the polynomial. To define a cyclic code, we pick a fixed polynomial, called generator polynomial. The codewords of this cyclic code are all the polynomials that are divisible by this generator polynomial.