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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.
As each primitive polynomial of degree 3 has three roots, all necessarily primitive, there are 12 / 3 = 4 primitive polynomials of degree 3. One primitive polynomial is x 3 + 2x + 1. Denoting one of its roots by γ, the algebraically conjugate elements are γ 3 and γ 9. The other primitive polynomials are associated with algebraically ...
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 multiplication uses one carryless multiply to produce a product (up to 2n − 1 bits), another carryless multiply of a pre-computed inverse of the field polynomial to produce a quotient = ⌊product / (field polynomial)⌋, a multiply of the quotient by the field polynomial, then an xor: result = product ⊕ ((field polynomial) ⌊product ...
A polynomial P with coefficients in a UFD R is then said to be primitive if the only elements of R that divide all coefficients of P at once are the invertible elements of R; i.e., the gcd of the coefficients is one. Primitivity statement: If R is a UFD, then the set of primitive polynomials in R[X] is closed under
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.
If g is a primitive root modulo p k, then g is also a primitive root modulo all smaller powers of p. If g is a primitive root modulo p k, then either g or g + p k (whichever one is odd) is a primitive root modulo 2 p k. [14] Finding primitive roots modulo p is also equivalent to finding the roots of the (p − 1)st cyclotomic polynomial modulo p.