# Arithmetic codes - Survey, recent and new results

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## Abstract

Arithmetic codes are used to check or correct arithmetic operations on computers, such as addition or modular addition, on integers which are represented in a fixed radix r (r≥2). The need for a measure of an error leads one to define the weight of an integer. Three definitions are known (one of them does not always satisfy the triangle inequality). From these notions, several problems arise (for instance, about perfect codes). We list these problems together with complete or partial previously known results, as well as some new results.

## Key words

AN-codes arithmetic codes completely regular codes perfect codes covering radius distance arithmetic distance external distance minimum distance modular distance dual degree radix-r form minimal form modified form nonadjacent form metric sphere triangle inequality weight arithmetic weight modular weight## Notations

- a|b
nonzero integer a divides integer b

- ℤ
_{m} ring of integers (mod m), ranging from 0 to m-1

- a≡b (mod c)
c|(b-a)

- |A|
cardinality of set A

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