Coding for the Lee and Manhattan Metrics with Weighing Matrices

TL;DR

This paper explores the use of weighing matrices, specifically conference and Hadamard matrices, to construct codes in Lee and Manhattan metrics, aiming to improve packing and error-correction capabilities.

Contribution

It introduces new code constructions using weighing matrices for Lee and Manhattan metrics, analyzing their properties for packing and transformation problems.

Findings

Weighing matrices serve as effective generator matrices for codes in Lee and Manhattan metrics.

Constructed codes demonstrate improved packing densities for Lee spheres.

Transformations preserve volume while inscribing Lee spheres in small cubes.

Abstract

This paper has two goals. The first one is to discuss good codes for packing problems in the Lee and Manhattan metrics. The second one is to consider weighing matrices for some of these coding problems. Weighing matrices were considered as building blocks for codes in the Hamming metric in various constructions. In this paper we will consider mainly two types of weighing matrices, namely conference matrices and Hadamard matrices, to construct codes in the Lee (and Manhattan) metric. We will show that these matrices have some desirable properties when considered as generator matrices for codes in these metrics. Two related packing problems will be considered. The first is to find good codes for error-correction (i.e. dense packings of Lee spheres). The second is to transform the space in a way that volumes are preserved and each Lee sphere (or conscribed cross-polytope), in the space, will be transformed to a shape inscribed in a small cube.