Communications Toolbox

Nonuniqueness of Representations

A given field has more than one primitive element. If two primitive elements have different minimal polynomials, then the corresponding matrices of elements will have their rows in a different order. If the two primitive elements share the same minimal polynomial, then the matrix of elements of the field is the same.

Note    You can use whatever primitive element you want, as long as you understand how the inputs and outputs of Galois field functions depend on the choice of some primitive polynomial. It is usually best to use the same primitive polynomial throughout a given script or function.

Other ways in which representations of elements are not unique arise from the equations that Galois field elements satisfy. For example, an exponential format of 8 in GF(9) is really the same as an exponential format of 0, because A⁸ = 1 = A⁰ in GF(9). As another example, the substitution mentioned just before the table Elements of GF(9) shows that the polynomial format [0 0 1] is really the same as the polynomial format [1 1].

List of All Elements of a Galois Field

Default Primitive Polynomials