same role among all polynomials as prime numbers among integers. Not all arbitrary irreducible polynomials need to be primitive, although in a special case p ¼ 2 and prime 2n 1 all irreducible polynomials are primitive. Proving the necessity and sufficiency of choosing feedback as indicated above would require some more algebra, which could take us far away from our main purpose. An interested reader may find the details in numerous sources (e.g. [31,32]).

Primitive polynomials are extensively tabulated in books on modern algebra and coding theory or (mostly for p ¼ 2) spread spectrum telecommunication [5,6,18,32]. Another option is a computer search, which is not at all a difficult task (e.g. Problem 6.47). In particular, ready-made functions for finding primitive polynomials are present in the Matlab Communications Toolbox.

As a whole, designing an m-sequence generator is pretty straightforward. When p is selected, a necessary length L determines the memory n, and finding an appropriate primitive polynomial exhausts the issue.

Commenting again on Examples 6.6.1 and 6.6.2, note that the binary m-sequence of length 7 is built based on the primitive polynomial f (x) ¼ x3 þ x þ 1 over GF(2), while the primitive polynomial over GF(3) used to generate the ternary sequence of length 26 is f (x) ¼ x3 þ 2x þ 1.

6.7 Periodic ACF of m-sequences

The results of the previous section lead quickly to the minimax binary sequences with ACF meeting (6.12). Consider a binary m-sequence fdig of memory n, i.e. length L ¼ 2n 1. Let us map its symbols 0, 1 onto a binary alphabet 1 according to the rule:

where in raising ( 1) to the degree di the latter is treated as though it is a real number 0 or 1. The sequence faig of real binary symbols 1 thus obtained has period N ¼ L ¼ 2n 1 and is a one-to-one image of the original binary m-sequence fdig. It is natural to keep for it the same name binary m-sequence as well. When the confusion is risky, a supplementary label like binary f 1g sequence versus binary f0, 1g sequence may be used. Let us find the non-normalized periodic ACF (6.7) of faig:

Now the shift-and-add property of binary f0, 1g m-sequences may be brought in. Addition in the exponent here may be treated as modulo 2, since it will produce the same result of exponentiation as an ordinary arithmetic summation. But then fd 0ig ¼ fdi þ di mg is a binary f0, 1g m-sequence of period L whenever m 6¼0modL,

6.8 More about finite fields

Let us take some element x of a finite field GF(p) and multiply it with itself m times, designating the result as the mth power of x:

x x . . . x ¼ xm

| {z }

m times

The ordinary rules of handling powers in conventional algebra remain valid in any field, including finite ones. In particular:

xmxn ¼ x x . . . x x x . . . x ¼ x x . . . x ¼ xmþn; ðxmÞn¼ xm xm . . . xm ¼ xmn

the inverse of any non-zero element give: x0 ¼ 1 and ðxmÞ 1¼ x m

Example 6.8.1. In the field GF(5) (see tables of Figure 6.11):

20 ¼ 1; 21 ¼ 2; 22 ¼ 2 2 ¼ 4; 23 ¼ 22 2 ¼ 4 2 ¼ 3; 24 ¼ 23 3 ¼ 3 2 ¼ 1 2 1 ¼ 3; 2 2 ¼ 2 1 2¼ 32 ¼ 4; 2 3 ¼ 2 1 3¼ 33 ¼ 4 3 ¼ 2; 2 4 ¼ 2 1 4¼ 34 ¼ 2 3 ¼ 1

Consider now successive degrees of the element x 6¼0 of GF(p): x0 ¼ 1, x1, x2, . . .. Since all terms of this series belong to GF(p), i.e. finite field, they cannot all be different, and therefore equality holds xi ¼ xk ) xi k ¼ 1 for some i > k. Suppose that the elementexists whose first p 1 powers 0 ¼ 1, 1, 2, . . . , p 2 are all different. Since p 1 is just the number of non-zero elements of GF(p), the powers above are exactly all non-zero elements of GF(p). Therefore, the element , if it really exists, allows constructing the whole field GF(p) but the zero element by just raising to powers 0, 1, . . . , p 2. Such an element is called a primitive one.

One of the most important facts about finite fields is that they all contain a primitive element. Proof of this result may be found in many algebraic or coding theory textbooks

(e.g. [30,32,33]). A primitive element is not unique: in any finite field whose order exceeds 3, more than one primitive element is present. For instance, as is seen from Example 6.8.1, both 2 and 3 are primitive elements in GF(5).

Since for a primitive element powers 0 ¼ 1, 1, 2, . . . , p 2 exhaust all non-zero elements of GF(p), p 1 should be equal to one of them. Actually it cannot be equal to anything but 1, because p 1 ¼ l with 0 < l p 2 means that p 1 l ¼ 1. This is not possible, since 1 < p 1 l < p 1 and among elements 1, 2, . . . , p 2 none may be equal to 1. Hence, p 1 ¼ 1. Now it is easy to see that the same is true for any non-zero element of a finite field, not only for a primitive one. Indeed, every non-zero element x of GF(p) is the lth power of a primitive element for a proper integer l: x ¼ l, so that (small Fermat theorem):

The next entity bears quite a natural name, fully consistent with the categories of ordinary algebra. The integer exponent l, which after raising to it produces x ¼ l, is the logarithm of x to the base with a conventional designation log x. Therefore,

log x ¼ x.

Clearly, the binary character is simply a mapping of the finite field GF(p) onto a pair of real numbers fþ1, 1g, transforming non-zero element x into þ1 if its logarithm is even and into 1 otherwise. Note that this mapping does not depend on a specific choice of a primitive element (Problem 6.24). The following properties of a binary character will be used further:

1. The character of the unit element of GF(p) is always one:

This is true because 0 ¼ 1 ) log 1 ¼ 0.

2.The character is a multiplicative function, i.e. the character of a product of two nonzero elements is a product of their characters. Indeed, from (6.17) and (6.18):

ðxyÞ ¼ ð 1Þlog ðxyÞ ¼ ð 1Þlog xþlog y ¼ ð 1Þlog x ð 1Þlog y ¼ ðxÞ ðyÞ ð6:20Þ

3. Balance property: the sum of characters of all non-zero elements of GF(p) is zero:

To prove this equality note that when x assumes all p 1 non-zero values, log x runs in some order over the range of p 1 integers 0, 1, . . . , p 2. Due to the oddness