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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 achieves high Hamming distance.
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.
A polynomial is primitive if its content equals 1. Thus the primitive part of a polynomial is a primitive polynomial. Gauss's lemma for polynomials states that the product of primitive polynomials (with coefficients in the same unique factorization domain) also is primitive. This implies that the content and the primitive part of the product of ...
If x is an algebraic number then a n x is an algebraic integer, where x satisfies a polynomial p(x) with integer coefficients and where a n x n is the highest-degree term of p(x). The value y = a n x is an algebraic integer because it is a root of q(y) = a n − 1 n p(y /a n), where q(y) is a monic polynomial with integer coefficients.
A monic irreducible polynomial of degree n having coefficients in the finite field GF(q), where q = p t for some prime p and positive integer t, is called a primitive polynomial if all of its roots are primitive elements of GF(q n). [2] [3] In the polynomial representation of the finite field, this implies that x is a primitive element.
The content of a polynomial p ∈ Z[X], denoted "cont(p)", is, up to its sign, the greatest common divisor of its coefficients. The primitive part of p is primpart(p) = p/cont(p), which is a primitive polynomial with integer coefficients. This defines a factorization of p into the product of an integer and a primitive polynomial. This ...
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.