On Lattice Packings and Coverings of Asymmetric Limited-Magnitude Balls

TL;DR

This paper develops lattice-based codes for asymmetric limited-magnitude error channels, achieving efficient packing and covering densities through novel constructions and mathematical techniques.

Contribution

It introduces new lattice constructions for error-correcting and covering codes tailored to limited-magnitude errors, with asymptotic optimality.

Findings

Some constructions attain constant asymptotic density

Utilizes diverse mathematical tools like Sidon sets and arithmetic progression avoidance

Provides lattice packings and coverings for multi-error limited-magnitude channels

Abstract

We construct integer error-correcting codes and covering codes for the limited-magnitude error channel with more than one error. The codes are lattices that pack or cover the space with the appropriate error ball. Some of the constructions attain an asymptotic packing/covering density that is constant. The results are obtained via various methods, including the use of codes in the Hamming metric, modular $B_t$-sequences, $2$-fold Sidon sets, and sets avoiding arithmetic progression.