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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.
Download QR code; Print/export Download as PDF; Printable version; In other projects ... In different branches of mathematics, primitive polynomial may refer to:
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.
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 ...
Download QR code; Print/export Download as PDF; Printable version; In other projects Wikimedia Commons; Wikiversity; ... Primitive polynomial (field theory)
Now, we can think of words as polynomials over , where the individual symbols of a word correspond to the different coefficients of the polynomial. To define a cyclic code, we pick a fixed polynomial, called generator polynomial. The codewords of this cyclic code are all the polynomials that are divisible by this generator polynomial.
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.