U. Glaeser

the product in GF(2 ν ), while a(x)b(x) denotes the regular product of polynomials. Notice that, for the
irreducible polynomial f(x), in particular, f(α) = 0 in GF(2 ν ), since f(x) � 0(mod f(x)).
So, the set GF(2 ν ) given by the irreducible polynomial f(x) of degree ν is the set of polynomials of
degree ≤ν − 1, where the sum operation is the regular sum of polynomials, and the product operation
is the residue of dividing by f(x) the regular product of two polynomials.
Example 5 Construct the field GF(8). Consider the polynomials of degree ≤2 over GF(2). Let f(x) = 1 +
x + x 3 . Since f(x) has no roots over GF(2), it is irreducible (notice that such an assessment can be made
only for polynomials of degree 2 or 3). Let us consider the powers of α modulo f(α). Notice that α 3 =
α 3 + f(α) = 1 + α. Also, α 4 = αα 3 = α (1 + α) = α + α 2 . Similarly, we obtain α 5 = αα 4 = α(α + α 2 ) =
α 2 + α 3 = 1 + α + α 2 , and α 6 = αα 5 = α + α 2 + α 3 = 1 + α 2 . Finally, α 7 = αα 6 = α + α 3 = 1.
Note that every nonzero element in GF(8) can be obtained as a power of the element α. In this case,
α is called a primitive element and the irreducible polynomial f(x) that defines the field is called a primitive
polynomial. It can be proven that it is always the case that the multiplicative group of a finite field is
cyclic, so there is always a primitive element.
A convenient description of GF(8) is given in Table 34.3. The first column in Table 34.3 describes
the element of the field in vector form, the second one as a polynomial in α of degree ≤2, the third one
as a power of α, and the last one gives the logarithm (also called Zech logarithm): it simply indicates
the corresponding power of α. As a convention, we denote by − ∞ the logarithm corresponding to the
element 0. �
It is often convenient to express the elements in a finite field as powers of α ; when we multiply two
of them, we obtain a new power of α whose exponent is the sum of the two exponents modulo 2 ν − 1.
Explicitly, if i and j are the logarithms of two elements in GF(2 ν ), then their product has logarithm i + j
(mod (2 ν − 1)). In the example above, if we want to multiply the vectors 101 and 111, we first look at
their logarithms. They are 6 and 5, respectively, so the logarithm of the product is 6 + 5(mod 7) = 4,
corresponding to the vector 011.
In order to add vectors, the best way is to express them in vector form and add coordinate to coordinate
in the usual way.
Cyclic Codes
In the same way we defined codes over the binary field GF(2), we can define codes over any finite field
GF(2 ν ). Now, a code of length n is a subset of (GF(2 ν )) n , but since we study only linear codes, we require
that such a subset is a vector space. Similarly, we define the minimum (Hamming) distance and the generator
and parity check matrices of a code. Some properties of binary linear codes, like the Singleton bound,
remain the same in the general case. Others, such as the Hamming bound, require some modifications.
Consider a linear code C over GF(2 ν ) of length n. We say that C is cyclic if, for any codeword (c 0, c 1, …,
c n−1) � C, then (c n−1, c 0, c 1,…, c n−2) � C. In other words, the code is invariant under cyclic shifts to the
right.
© 2002 by CRC Press LLC
TABLE 34.3 The Finite Field GF(8) Generated by 1 + x + x 3
Vector Polynomial Power of α Logarithm
000 0 0 −∞
100 1 1 0
010 α α 1
001 α 2
α 2
2
110 1 + α α 3
3
011 α + α 2
α 4
4
111 1 + α + α 2
α 5
5
101 1 + α 2
α 6
6