2022/07/05 by Clementa Alonso-González, Alonso-González, Clementa, Miguel Ángel Navarro-Pérez +1
Computer Science · Engineering · #Cooperative Communication and Network Coding #Cellular Automata and Applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2207.01997
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 \mathbbFq. If the list of dimensions is (1, …, 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 \mathbbFqn 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.