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In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(p m).This means that a polynomial F(X) of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(p m) such that {,,,,, …} is the entire field GF(p m).
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.
Linear block code Polynomial code Cyclic code BCH code Reed–Solomon code: Block length: n = q − 1: Message length: k: Distance: n − k + 1: Alphabet size: q = p m (p prime) Notation [n, k, n − k + 1] q-code: Algorithms; Berlekamp–Massey Euclidean et al. Properties; Maximum-distance separable code
Download QR code; Print/export Download as PDF; Printable version; In other projects ... In different branches of mathematics, primitive polynomial may refer to:
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.
It follows that composition induces a well-defined operation on primitive classes of discriminant , and as mentioned above, Gauss showed these classes form a finite abelian group. The identity class in the group is the unique class containing all forms x 2 + B x y + C y 2 {\displaystyle x^{2}+Bxy+Cy^{2}} , i.e., with first coefficient 1.
By 1963 (or possibly earlier), J. J. Stone (and others) recognized that Reed–Solomon codes could use the BCH scheme of using a fixed generator polynomial, making such codes a special class of BCH codes, [4] but Reed–Solomon codes based on the original encoding scheme are not a class of BCH codes, and depending on the set of evaluation ...