Optimum Distance Flag Codes from Spreads via Perfect Matchings in Graphs

TL;DR

This paper characterizes and constructs optimum distance flag codes in finite vector spaces using graph perfect matchings, achieving maximum distance and size for given type vectors.

Contribution

It provides a characterization of admissible type vectors and a novel construction method for optimum distance flag codes based on perfect matchings in graphs.

Findings

Characterization of admissible type vectors for optimum flag codes

Construction method using perfect matchings in graphs

Achieves maximum distance and largest size for given type vectors

Abstract

In this paper, we study flag codes on the vector space $\mathbb{F}_q^n$, being $q$ a prime power and $\mathbb{F}_q$ the finite field of $q$ elements. More precisely, we focus on flag codes that attain the maximum possible distance (optimum distance flag codes) and can be obtained from a spread of $\mathbb{F}_q^n$. We characterize the set of admissible type vectors for this family of flag codes and also provide a construction of them based on well-known results about perfect matchings in graphs. This construction attains both the maximum distance for its type vector and the largest possible cardinality for that distance.