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Building on the GF(8) example above, this section explains the interpretation of array elements in a Galois array in greater generality. The field GF(2^m) has 2^m distinct elements, which this toolbox labels as 0, 1, 2,..., 2^m-1. These integer labels correspond to elements of the Galois field via a polynomial expression involving a primitive element of the field. More specifically, each integer between 0 and 2^m-1 has a binary representation in m bits. Using the bits in the binary representation as coefficients in a polynomial, where the least significant bit is the constant term, leads to a binary polynomial whose order is at most m-1. Evaluating the binary polynomial at a primitive element of GF(2^m) leads to an element of the field.
Conversely, any element of GF(2^m) can be expressed as a binary polynomial of order at most m-1, evaluated at a primitive element of the field. The m-tuple of coefficients of the polynomial corresponds to the binary representation of an integer between 0 and 2^m.
Below is a symbolic illustration of the correspondence of an integer X, representable in binary form, with a Galois field element. Each bk is either zero or one, while A is a primitive element.
| | Example: Representing Elements of GF(8) | Example: Representing a Primitive Element | |
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