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In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
Algorithm: SFF (Square-Free Factorization) Input: A monic polynomial f in F q [x] where q = p m Output: Square-free factorization of f R ← 1 # Make w be the product (without multiplicity) of all factors of f that have # multiplicity not divisible by p c ← gcd(f, f′) w ← f/c # Step 1: Identify all factors in w i ← 1 while w ≠ 1 do y ...
The polynomial x 2 + 2x + 2, on the other hand, is primitive. Denote one of its roots by α. Then, because the natural numbers less than and relatively prime to 3 2 − 1 = 8 are 1, 3, 5, and 7, the four primitive roots in GF(3 2) are α, α 3 = 2α + 1, α 5 = 2α, and α 7 = α + 2. The primitive roots α and α 3 are algebraically conjugate.
This may be verified by factoring X 64 − X over GF(2). The elements of GF(64) are primitive n {\displaystyle n} th roots of unity for some n {\displaystyle n} dividing 63 {\displaystyle 63} . As the 3rd and the 7th roots of unity belong to GF(4) and GF(8) , respectively, the 54 generators are primitive n th roots of unity for some n in {9, 21 ...
As the integrand is the third-degree polynomial y(x) = 7x 3 – 8x 2 – 3x + 3, the 2-point Gaussian quadrature rule even returns an exact result. In numerical analysis , an n -point Gaussian quadrature rule , named after Carl Friedrich Gauss , [ 1 ] is a quadrature rule constructed to yield an exact result for polynomials of degree 2 n − 1 ...
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials.The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong.
The generating function F for this transformation is of the third kind, = (,). To find F explicitly, use the equation for its derivative from the table above, =, and substitute the expression for P from equation , expressed in terms of p and Q:
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.