Search results
Results From The WOW.Com Content Network
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.
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 different branches of mathematics, primitive polynomial may refer to: Primitive polynomial (field theory) , a minimal polynomial of an extension of finite fields Primitive polynomial (ring theory) , a polynomial with coprime coefficients
The polynomials with the largest order are called primitive polynomials, and for polynomials of degree with binary coefficients, have order . All errors in an odd number of bits will be detected by a polynomial which is a multiple of x + 1 {\displaystyle x+1} .
Getty Images Detroit slang is an ever-evolving dictionary of words and phrases with roots in regional Michigan, the Motown music scene, African-American communities and drug culture, among others.
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.