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The generator polynomial of the BCH code is defined as the least common multiple g(x) = lcm(m 1 (x),…,m d − 1 (x)). It can be seen that g(x) is a polynomial with coefficients in GF(q) and divides x n − 1. Therefore, the polynomial code defined by g(x) is a cyclic code.
Here are some examples of such properties: A polynomial code is cyclic if and only if the generator polynomial divides . If the generator polynomial is primitive, then the resulting code has Hamming distance at least 3, provided that . In BCH codes, the generator polynomial is chosen to have specific roots in an extension field, in a way that ...
Over GF(2), x + 1 is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by x + 1 (it has 1 as a root). An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n ...
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 ...
A 16-bit Galois LFSR. The register numbers above correspond to the same primitive polynomial as the Fibonacci example but are counted in reverse to the shifting direction. This register also cycles through the maximal number of 65535 states excluding the all-zeroes state. The state ACE1 hex shown will be followed by E270 hex.
Its generator polynomial as a cyclic code is given by f ( x ) = ∏ j ∈ Q ( x − ζ j ) {\displaystyle f(x)=\prod _{j\in Q}(x-\zeta ^{j})} where Q {\displaystyle Q} is the set of quadratic residues of p {\displaystyle p} in the set { 1 , 2 , … , p − 1 } {\displaystyle \{1,2,\ldots ,p-1\}} and ζ {\displaystyle \zeta } is a primitive p ...
The examples above discuss the representation problem for the numbers 3 and 65 by the form + and for the number 1 by the form . We see that 65 is represented by x 2 + y 2 {\displaystyle x^{2}+y^{2}} in sixteen different ways, while 1 is represented by x 2 − 2 y 2 {\displaystyle x^{2}-2y^{2}} in infinitely many ways and 3 is not represented by ...
The polynomials with the largest order are called primitive polynomials, and for polynomials of degree with binary coefficients, have order . All errors in an odd number of bits will be detected by a polynomial which is a multiple of +. This is equivalent to the polynomial having an even number of terms with non-zero coefficients.