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This chapter investigates the properties of (linear) codes in $ A_n $lattices, the practical motivation for which is found in several communicationscenarios, such as asymmetric channels, sticky-insertion channels, bit-shiftchannels, and permutation channels. In particular, a connection between thesecodes and notions of difference sets and Sidon sets in Abelian groups isdemonstrated. It is shown that the $ A_n $ lattice admits a linear perfect codeof radius $ 1 $ if and only if there exists an Abelian planar difference set ofcardinality $ n + 1 $. Similarly, a direct link is given between linear codesof radius $ r $ in the $ A_n $ lattice and Sidon sets of order $ 2r $ andcardinality $ n + 1 $. Sidon sets of order $ 2r-1 $ are also representedgeometrically in a similar way. Apart from providing geometric intuition aboutSidon sets, this interpretation enables simple derivations of bounds on theirparameters, which are either equivalent to, or improve upon the known bounds.In connection to the above, more general (non-planar) Abelian difference setsand perfect codes of radius $ r $ are also discussed.

Sidon Sets, Difference Sets, and Codes in $ A_n $ Lattices