Lower bounds on the minimum distance of long codes in the Lee metric

TL;DR

This paper reviews Gilbert bounds for Lee metric codes and introduces new constructive lower bounds using geometric codes and concatenation techniques, improving understanding of minimum Lee distances for various alphabets.

Contribution

It provides new lower bounds on the minimum Lee distance of long codes, surpassing existing existential bounds through geometric and concatenation methods.

Findings

Derived lower bounds using algebraic geometric arguments.

Constructed bounds via concatenation with Hamming metric codes.

Improved minimum distance estimates for Lee metric codes.

Abstract

The Gilbert type bound for codes in the title is reviewed, both for small and large alphabets. Constructive lower bounds better than these existential bounds are derived from geometric codes, either over Fp or Fp2 ; or over even degree extensions of Fp: In the latter case the approach is concatena- tion with a good code for the Hamming metric as outer code and a short code for the Lee metric as an inner code. In the former case lower bounds on the minimum Lee distance are derived by algebraic geometric arguments inspired by results of Wu, Kuijper, Udaya (2007).