Coding & GF(2): Understanding Logic in a Field

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However, the professor is using it as a shorthand for the polynomial 1+x, which is the inverse of x+1.In summary, in the context of GF(2), the polynomials 3xy^5 and \frac{1}{2}x^2y can be reduced to 3xy and xy, respectively. This is because y^5 can be simplified to y, x^2 is equivalent to x, and 1/2 represents the polynomial 1+x, which is the inverse of x+1.
  • #1
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This came up in my logic course.

The professor writes that in GF(2), the polynomials [tex]3xy^5[/tex] and [tex]\frac{1}{2}x^2y[/tex] respectively can be reduced to [tex]3xy[/tex] and [tex]xy[/tex].

I understand that [tex]y^5=(y^2)(y^2)y=(1)(1)y=y[/tex], but also in GF(2), for any x, we have x+x=0. So it seems to me that [tex]3xy^5[/tex] can be further reduced to just [tex]xy[/tex] because we have [tex]3xy=xy+xy+xy=0+xy=xy[/tex].

For the other, I am clueless. Sure, [tex]x^2y=xy[/tex] because in GF(2), for any x, we have x²=x. But what happened to the 1/2? Worse, what is 1/2? It's the inverse of 2. And 2 is 1+1. But 1+1=0, and 0 has no inverse in a field.

What's going on here?
 
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First off, just to make sure it's clear, we're talking about polynomial functions, not polynomials. The distinction is subtle, yet vital.

Secondly, [itex]x^2 \not\equiv 1[/itex]; I'm not sure where you got that idea. The right fact is [itex]x^2 \equiv x[/itex].

Thirdly, 1/2 doesn't make any sense in GF(2), because as you say, it's equivalent to 1/0.
 
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In GF(2), the elements are restricted to only 0 and 1. This means that any other number, such as 1/2, is equivalent to 0. Therefore, in GF(2), 1/2 is the same as 0 and does not have an inverse. This is why the polynomial \frac{1}{2}x^2y can be reduced to just xy, as the 1/2 is equivalent to 0 and has no effect on the polynomial.

Additionally, in GF(2), the addition operation is equivalent to the XOR operation. This means that when we add two elements, we are essentially performing a binary addition and carrying over any remainders. In the case of 3xy^5, we can see that it can be reduced to just xy because 3 is equivalent to 1 in GF(2) and the addition of two 1s results in 0.

Overall, understanding logic in a field, specifically GF(2), involves understanding the restrictions and operations within that field. In this case, we see that elements like 1/2 and 3 behave differently in GF(2) compared to traditional arithmetic, and understanding these differences is crucial in effectively working with polynomials in GF(2).
 

1. What is coding in the context of GF(2)?

Coding in GF(2) refers to the process of representing and manipulating digital data using elements of the Galois Field of order 2. This field is commonly used in computer science and engineering for its applications in error correction and cryptography.

2. How does logic work in GF(2)?

In GF(2), logic works by representing boolean operations (AND, OR, NOT) as arithmetic operations on elements of the field. For example, the boolean operation of AND can be represented as multiplication in GF(2), where 0 represents False and 1 represents True.

3. What are the benefits of using GF(2) in coding?

There are several benefits of using GF(2) in coding, including its ability to detect and correct errors in digital data through the use of error-correcting codes. GF(2) is also useful in cryptography, as it provides a finite set of elements that can be used for secure encryption and decryption.

4. Can GF(2) be used to code all types of data?

No, GF(2) is primarily used for coding digital data, such as binary data. It is not typically used for coding analog data, as it is not a continuous field.

5. How can understanding GF(2) logic benefit me as a programmer?

Understanding GF(2) logic can benefit you as a programmer by providing a deeper understanding of how digital data is represented and manipulated in computer systems. It can also be useful in developing efficient and secure coding techniques for error correction and cryptography.

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