The packing radius of a code and partitioning problems: The case for poset metrics on finite vector spaces

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 well known 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, which we will extend 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.