Flag Codes: Distance Vectors and Cardinality Bounds

TL;DR

This paper introduces the concept of distance vectors for flag codes over finite fields, providing a detailed structural analysis and bounds on their maximum size given fixed parameters.

Contribution

It introduces distance vectors as a new algebraic tool to analyze flag codes, enabling a more detailed understanding and bounds on their sizes.

Findings

Distance vectors encode more information than flag distances.

Structural characterization of flag codes using distance vectors.

Bounds on maximum size of flag codes with fixed minimum distance.

Abstract

Given $\mathbb{F}_q$ the finite field with $q$ elements and an integer $n\geq 2$, a flag is a sequence of nested subspaces of $\mathbb{F}_q^n$ and a flag code is a nonempty set of flags. In this context, the distance between flags is the sum of the corresponding subspace distances. Hence, a given flag distance value might be obtained by many different combinations. To capture such a variability, in the paper at hand, we introduce the notion of distance vector as an algebraic object intrinsically associated to a flag code that encloses much more information than the distance parameter itself. Our study of the flag distance by using this new tool allows us to provide a fine description of the structure of flag codes as well as to derive bounds for their maximum possible size once the minimum distance and dimensions are fixed.