Motzkin numbers and flag codes

TL;DR

This paper reveals a novel connection between Motzkin numbers and the enumeration of distance vectors in full flag codes within network coding, highlighting a combinatorial structure underlying these codes.

Contribution

It establishes that the number of distance vectors for full flag codes corresponds exactly to Motzkin numbers, introducing a new combinatorial perspective in network coding.

Findings

Number of distance vectors equals the n-th Motzkin number.

Identifies the sequence counting distance vectors with prescribed minimum distance.

Provides a combinatorial interpretation of flag codes in terms of Motzkin numbers.

Abstract

Motzkin numbers have been widely studied since they count many different combinatorial objects. In this paper we present a new appearance of this remarkable sequence in the network coding setting through a particular case of multishot codes called flag codes. A flag code is a set of sequences of nested subspaces (flags) of a vector space over the finite field $\mathbb{F}_q$. If the list of dimensions is $(1, \dots, n-1)$, we speak about a full flag code. The flag distance is defined as the sum of the respective subspace distances and can be represented by means of the so-called distance vectors. We show that the number of distance vectors corresponding to the full flag variety on $\mathbb{F}_q^n$ is exactly the $n$-th Motzkin number. Moreover, we can identify the integer sequence that counts the number of possible distance vectors associated to a full flag code with prescribed minimum distance.