# Primitive polynomial in GF(4) ?

• mahesh_2961
In summary, the conversation discusses the use of a primitive polynomial in Galois Field(4) to generate a PN sequence of length 1023 for a pseudo random generator. The speaker is unsure if a primitive polynomial of order >5 exists in GF(4) and is seeking information on this topic.
mahesh_2961
primitive polynomial in GF(4) ??

Hai All,
I m required to make a pseudo random generator. I know that i can make that using some Flip flops and XOR gates(in linear feedback shift register configuration ). But the resulting PN sequences will be in Galois Field(2) as the taps for the flip flops are according to the coefficients of a primitive polynomial in GF(2).

So i was just thinking if there is any primitive polynomial in GF(4)? I need to make a pn sequence of length 1023. In GF(2), the tap weights for the linear feedback shift register is the coefficient of primitive polynomial of order 10 . So in GF(4), if a primitive polynomial exists, a primitive polynomial of order 5 will give taps for an LFSR to generate PNsequence of length 1023 .
Is there a primitive polynomial of order >5 in GF(4) ?

i searched the net but couldn't get any information regarding this ..

Mahesh

E.g. take the ##n##-th root of unity, where ##n## is odd and consider its irreducible polynomial over ##GF(4)##.

## 1. What is a primitive polynomial in GF(4)?

A primitive polynomial in GF(4) is a polynomial with coefficients in the Galois Field GF(4) that generates all nonzero elements in the field when raised to different powers.

## 2. How is a primitive polynomial in GF(4) different from a primitive polynomial in GF(2)?

A primitive polynomial in GF(4) is different from a primitive polynomial in GF(2) in terms of the field they are defined in. GF(4) is a finite field with 4 elements, while GF(2) is a finite field with 2 elements. This affects the structure and properties of the polynomials in these fields.

## 3. How do you determine if a polynomial is primitive in GF(4)?

To determine if a polynomial is primitive in GF(4), you can use the definition of primitive polynomial and check if it generates all nonzero elements in the field. This can also be done by calculating the order of the polynomial and checking if it is equal to the order of the field, which is 3.

## 4. Can a polynomial be primitive in both GF(4) and GF(2)?

No, a polynomial cannot be primitive in both GF(4) and GF(2) as they are defined in different fields and have different properties. A polynomial can be primitive in one field but not in the other.

## 5. What are some real-world applications of primitive polynomials in GF(4)?

Primitive polynomials in GF(4) have applications in coding theory, cryptography, and error-correction techniques. They are also used in the design of digital communication systems and data encryption algorithms.

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