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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 ...
In this case, a primitive element is also called a primitive root modulo q. For example, 2 is a primitive element of the field GF(3) and GF(5), but not of GF(7) since it generates the cyclic subgroup {2, 4, 1} of order 3; however, 3 is a primitive element of GF(7). The minimal polynomial of a primitive element is a primitive polynomial.
That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n. So g is a primitive root modulo n if and only if g is a generator of the multiplicative group of integers modulo n.
The generator polynomial of the BCH code is defined as the least common multiple () ... is called primitive. The generator polynomial () of a BCH code has ...
[2] [3] In the polynomial representation of the finite field, this implies that x is a primitive element. There is at least one irreducible polynomial for which x is a primitive element. [4] In other words, for a primitive polynomial, the powers of x generate every nonzero value in the field.
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.
The polynomial factors into linear factors over a field of order q. More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order q. This implies that, if q = p n then X q − X is the product of all monic irreducible polynomials over GF(p), whose degree divides n.
If is not injective, let p(X) be a generator of its kernel, which is thus the minimal polynomial of θ. The image of φ {\displaystyle \varphi } is a subring of L , and thus an integral domain . This implies that p is an irreducible polynomial, and thus that the quotient ring K [ X ] / p ( X ) {\displaystyle K[X]/\langle p(X)\rangle } is a field.