The Packing Radius of a Code and Partitioning Problems: the Case for Poset Metrics

TL;DR

This paper extends the understanding of the packing radius for poset codes to general posets, linking it to a generalized partition problem and developing algorithms for its computation.

Contribution

It introduces a novel approach to determine the packing radius of any poset code by reducing the problem to a generalized partition problem and extending existing algorithms.

Findings

The packing radius problem is equivalent to a generalized partition problem.

Extended the differentiating method to solve the generalized partition problem.

Developed methods to compare packing radii without explicit calculation.

Abstract

Until this work, the packing radius of a poset code was only known in the cases where the poset was a chain, a hierarchy, a union of disjoint chains of the same size, and for some families of codes. Our objective is to approach the general case of any poset. To do this, we will divide the problem into two parts. The first part consists in finding the packing radius of a single vector. We will show that this is equivalent to a generalization of a famous NP-hard problem known as "the partition problem". Then, we will review the main results known about this problem giving special attention to the algorithms to solve it. The main ingredient to these algorithms is what is known as the differentiating method, and therefore, we will extend it to the general case. The second part consists in finding the vector that determines the packing radius of the code. For this, we will show how it is sometimes possible to compare the packing radius of two vectors without calculating them explicitly.