Search results
Results From The WOW.Com Content Network
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 ...
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.
A primitive polynomial of degree m has m different roots in GF(p m), which all have order p m − 1, meaning that any of them generates the multiplicative group of the field. Over GF( p ) there are exactly φ ( p m − 1) primitive elements and φ ( p m − 1) / m primitive polynomials, each of degree m , where φ is Euler's totient function .
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 ...
Download QR code; Print/export Download as PDF; Printable version ... In different branches of mathematics, primitive polynomial may refer to: Primitive polynomial ...
Otherwise, θ is algebraic over K; that is, θ is a root of a polynomial over K. The monic polynomial of minimal degree n, with θ as a root, is called the minimal polynomial of θ. Its degree equals the degree of the field extension, that is, the dimension of L viewed as a K-vector space.
A form is primitive if its content is 1, that is, if its coefficients are coprime. If a form's discriminant is a fundamental discriminant , then the form is primitive. [ 1 ] Discriminants satisfy Δ ≡ 0 , 1 ( mod 4 ) . {\displaystyle \Delta \equiv 0,1{\pmod {4}}.}
The Conway polynomial C p,n is defined as the lexicographically minimal monic primitive polynomial of degree n over F p that is compatible with C p,m for all m dividing n. This is an inductive definition on n: the base case is C p,1 (x) = x − α where α is the lexicographically minimal primitive element of F p. The notion of lexicographical ...