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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 x 1 {\displaystyle x^{1}} to get x 3 + x 2 + x {\displaystyle x^{3}+x^{2}+x} .
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 ...
For dense polynomials, such as the CRC-32 polynomial, computing the remainder a byte at a time produces equations where each bit depends on up to 8 bits of the previous iteration. In byte-parallel hardware implementations this calls for either 8-input or cascaded XOR gates which have substantial gate delay.
Code words look like polynomials. By design, the generator polynomial has consecutive roots ... Where S(x) is the partial syndrome polynomial: [4] () ...
A general theorem of Matiyasevich says that if a set is defined by a system of Diophantine equations, it can also be defined by a system of Diophantine equations in only 9 variables. [8] Hence, there is a prime-generating polynomial inequality as above with only 10 variables. However, its degree is large (in the order of 10 45). On the other ...
The elements of an algebraic number field are usually represented as polynomials in a generator of the field which satisfies some univariate polynomial equation. To work with a polynomial system whose coefficients belong to a number field, it suffices to consider this generator as a new variable and to add the equation of the generator to the ...
For example, if the taps are at the 16th, 14th, 13th and 11th bits (as shown), the feedback polynomial is + + + + The "one" in the polynomial does not correspond to a tap – it corresponds to the input to the first bit (i.e. x 0, which is equivalent to 1). The powers of the terms represent the tapped bits, counting from the left.
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
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