# Separate arithmetic codes closed under addition

For error detection purpose I am looking for separate arithmetic codes which are closed under integer addition.

By separate, I mean the code word $C$ for message $x$ is a tuple $(x,f(x))$ where $f(x)$ is the redundancy part. In contrast to that, a non-separate code word $C$ would be $f_2(x)$ where $f_2$ encodes the word to a redundant representation such as e.g. AN-codes.

So far, I know (multi) residue codes. I found another form of residue codes in http://mark.bu.edu/papers/192.pdf where $f(x) = x^2 \text{ mod } p$ but I haven't found any others. Can someone point me to information to other codes of that type?

• This is right, but linear codes are not arithmetic codes and not closed under integer addition. – Peter W. Feb 22 '17 at 14:07
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