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Math Help - what is Galois Field

  1. #1
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    what is Galois Field

    I'm reading about reed-Solomon error correction. They venture into Galois fields. I am not from mathematics background. I tried to Google around but I can't get my head around these Galois fields. Can anybody explain to me in simple form these Galois fields?
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  2. #2
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    Re: what is Galois Field

    A field is a mathematical structure with a lot of useful properties. In general, you can take an arbitrary set and define two binary operators on it. If the set and binary operators satisfy some rather strict requirements, the structure you created is a field. A Galois Field is a specific finite field with specific binary operators that very much resemble addition and multiplication that you are used to, but with some distinct differences. Each Galois Field has a certain size. For example, GF(5) is typically represented with the elements \{0,1,2,3,4\} and the binary operators are modular addition and modular multiplication. However, that is because that set equipped with modular addition and modular multiplication is isomorphic to GF(5). Each Galois Field is unique in the sense that any other field of the same size is isomorphic to it. In other words, your field could have elements in it that look nothing like numbers. Your binary operators might not make any sense when related to numbers. But, there is a way to assign numbers to each element of your set so that the binary operators become very similar to addition and multiplication.
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    Re: what is Galois Field

    thanks for the answer SlipE. I can see the small light coming now. 4+4 = 3 (mod 5), 4x3 = 2 (mod 5) right?... By properties I guess you mean the famous axioms - associativity, commutativity, etc. but now what is isomorphic?
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  4. #4
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    Re: what is Galois Field

    Isomorphisms are a little tricky to define without more mathematical knowledge. To illustrate the idea, I will use a very small example. Suppose you have a set A = \{a,b\} and define operations \oplus: A \times A \to A, \otimes: A \times A \to A (that means that each operator takes two elements of A and gives back one element of A) by

    \begin{align*}a\oplus a & = a \\ a \oplus b & = b \\ b \oplus a & = b \\ b\oplus b & = a \\ a\otimes a & = a \\ a \otimes b & = a \\ b \otimes a & = a \\ b \otimes b & = b\end{align*}

    Then if you map a \mapsto 0, b \mapsto 1 and use modular addition and multiplication (mod 2), then map back 0 \mapsto a, 1 \mapsto b, you get the same result as if you had done the corresponding operations in A. In general, two fields are isomorphic if there is a bijective mapping between them that "preserves their structure". There are too many details to convey in a short post the specifics of how the structure is actually preserved, so I hope the example I provided gives a little insight.
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    Re: what is Galois Field

    Thanks a lot SlipE.
    Things are starting to make sense bit by bit now. As I continue with my R-S reading, I see they are using primitive polynomials to map elements of GF(2^m) to binary values. So I'll like to guess that's where Isomorphisms comes into play.
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