Recognizing and Realizing Inductively Pierced Codes

TL;DR

This paper characterizes inductively pierced codes algebraically and combinatorially, provides a polynomial-time method to find piercing orders, and constructs realizations with open balls, also determining minimal realization dimensions.

Contribution

It offers the first algebraic and combinatorial characterizations of inductively pierced codes and efficient algorithms for their realization.

Findings

Polynomial-time algorithm for computing piercing orders.

Algebraic and combinatorial characterizations of inductively pierced codes.

Classification of minimal realization dimensions.

Abstract

We prove algebraic and combinatorial characterizations of the class of inductively pierced codes, resolving a conjecture of Gross, Obatake, and Youngs. Starting from an algebraic invariant of a code called its canonical form, we explain how to compute a piercing order in polynomial time, if one exists. Given a piercing order of a code, we explain how to construct a realization of the code using a well-formed collection of open balls, and classify the minimal dimension in which such a realization exists.