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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 ...
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.
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 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.
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.
Download QR code; Print/export Download as PDF; Printable version; In other projects ... In different branches of mathematics, primitive polynomial may refer to:
Download QR code; Print/export Download as PDF; Printable version; In other projects ... 18,407,808 = number of primitive polynomials of degree 29 over GF(2) [14]
Its generator polynomial as a cyclic code is given by f ( x ) = ∏ j ∈ Q ( x − ζ j ) {\displaystyle f(x)=\prod _{j\in Q}(x-\zeta ^{j})} where Q {\displaystyle Q} is the set of quadratic residues of p {\displaystyle p} in the set { 1 , 2 , … , p − 1 } {\displaystyle \{1,2,\ldots ,p-1\}} and ζ {\displaystyle \zeta } is a primitive p ...