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

TL;DR

This paper explores the relationship between codes in $A_n$ lattices, difference sets, and Sidon sets in Abelian groups, providing geometric insights and bounds relevant to communication channels.

Contribution

It establishes new connections between lattice codes, difference sets, and Sidon sets, and offers geometric interpretations that lead to improved bounds on their parameters.

Findings

A linear perfect code of radius 1 exists in $A_n$ if and only if an Abelian planar difference set of size $n+1$ exists.

Linear codes of radius $r$ correspond to Sidon sets of order $2r$ with size $n+1$.

Geometric interpretation yields new bounds on Sidon set parameters.

Abstract

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