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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 ...
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 ...
Download QR code; Print/export Download as PDF; Printable version; ... In different branches of mathematics, primitive polynomial may refer to: Primitive polynomial ...
GF(2) (also denoted , Z/2Z or /) is the finite field with two elements. [1] [a]GF(2) is the field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively 0 and 1, as usual.
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 ...
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 ...
In the following examples it is best not to use the polynomial representation, as the meaning of x changes between the examples. The monic irreducible polynomial x 8 + x 4 + x 3 + x + 1 over GF(2) is not primitive. Let λ be a root of this polynomial (in the polynomial representation this would be x), that is, λ 8 + λ 4 + λ 3 + λ + 1 = 0.