Optimum Distance Flag Codes from Spreads via Perfect Matchings in Graphs

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.