Geometric representations of binary codes embeddable in three dimensions

TL;DR

This paper explores geometric representations of binary linear codes, showing all can be embedded in four dimensions and characterizing those embeddable in three dimensions via graph cut spaces, with polynomial algorithms for minimal dimension.

Contribution

It establishes that every binary linear code has a geometric representation in R^4 and characterizes those in R^3 as cut spaces of graphs, providing a polynomial-time decision method.

Findings

All binary linear codes have geometric representations in R^4.

A binary code has a geometric representation in R^3 iff it is a graph cut space.

Deciding the minimal embedding dimension is polynomial-time computable.

Abstract

We say that a binary linear code C has a geometric representation if there exists a two dimensional simplicial complex D such that C is a punctured code of the kernel ker D of the incidence matrix of D and dim C = dim ker D. We show that every binary linear code has a geometric representation that can be embedded into R^4. Moreover, we show that a binary linear code C has a geometric representation in R^3 if and only if there exists a graph G such that C equals the cut space of G. This is a polynomially testable property and hence we can conclude that there is a polynomial algorithm that decides the minimal dimension of a geometric representation of a binary linear code.