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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.
Bahasa Indonesia; Italiano; ... Download QR code; Print/export Download as PDF; ... Primitive polynomial (field theory) Principal equation form;
Download QR code; Print/export Download as PDF; Printable version; In other projects ... In different branches of mathematics, primitive polynomial may refer to:
Bahasa Indonesia; Slovenčina; ... Download QR code; Print/export Download as PDF; ... Square-free integer; Square-free polynomial; Square number;
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
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.