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I am currently reading about RS codes. I see that they are using a Galois Fields (Finite Fields) as vector spaces. Is there any other particular reason other than the fact that they simplify binary arithmetic and for example in $GF(2^8)$ each byte can be considered as a vector? Can they work in vector spaces that are defined on infinite fields like $\mathbb{Q}$. Thanks in advance for your time.

PS: Sorry in advance if this isn't the right place to post this question, but I saw that both Math and Crypto StachExchanges have coding-theory tag.

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    $\begingroup$ In crypto, we generally don't work in $\mathbb{Q}$; for boring practical reasons, we prefer messages that can be expressed in a bounded number of bits. $\endgroup$
    – poncho
    Feb 22, 2022 at 20:28
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    $\begingroup$ We also like to be able to define a uniform probability distribution over a space! $\endgroup$
    – Mikero
    Feb 22, 2022 at 21:24
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    $\begingroup$ What poncho and Mikero mention makes sense, and these are crucial reasons why we don't consider infinite algebraic structures in cryptography. However, just to satisfy your curiosity: Reed-Solomon codes can be easily applied over any field, no matter what size. In fact, they exist over any ring, no matter the size, as long as it contains a large enough "exceptional sequence" (e.g. crypto.stackexchange.com/a/96507/13843). $\endgroup$
    – Daniel
    Feb 22, 2022 at 21:55
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    $\begingroup$ However, Reed-Solomon codes on their own simply take a message and add some redundancy for decoding errors. Their use in cryptography, for example in Shamir secret-sharing, requires sampling uniformly random elements over this structure, which as Mikero mentioned is not possible. $\endgroup$
    – Daniel
    Feb 22, 2022 at 21:55

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Yes they can work, and under some channel noise conditions be useful for error correction coding in a continuous channel. This idea was originally due to Prof. Welch (of Welch-Berlekamp algorithm and Welch bound fame) who had unpublished lecture notes about it in the 1980s, and from an engineering point of view $\mathbb{C}$ was the obvious field to use, where the issue of existence of primitive roots of unity of any desired order $n$ is trivial, just take $\omega=\exp\{2 \pi i/n\}.$

As the comments pointed out, this is not so useful for cryptography, since the existence of uniform distributions are crucial for certain protocols. Of course, Reed-Solomon codes in their field evaluation formulation are intimately linked to Shamir secret sharing, say with threshold $t,$ but in a finite field setting for enabling no leakage of information if less than $t$ shares are known.

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