Skeleton Simplicial Evaluation Codes

TL;DR

This paper investigates evaluation codes derived from subspace arrangements over finite fields, linking code parameters to combinatorial properties of associated simplicial complexes, and determines minimum distances for specific skeleton arrangements.

Contribution

It introduces a new connection between evaluation codes and the combinatorics of simplicial complexes, providing bounds and exact parameters for certain arrangements.

Findings

Code length equals the characteristic polynomial of the subspace arrangement.

Dimension bounds relate to the face vector of the simplicial complex.

Minimum distance is explicitly determined for skeleton arrangements.

Abstract

For a subspace arrangement over a finite field we study the evaluation code defined on the arrangement's set of points. The length of this code is given by the subspace arrangement's characteristic polynomial. For coordinate subspace arrangements, the dimension is bounded below by the face vector of the corresponding simplicial complex. The minimum distance is determined for coordinate subspace arrangements where the simplicial complex is a skeleton.