Cardinality-consistent flag codes with longer type vectors

TL;DR

This paper introduces new constructions of flag codes with longer type vectors on finite fields, achieving optimal distances and cardinalities by uniting cyclic orbit structures.

Contribution

It presents a comprehensive construction uniting cyclic orbit flag codes, resulting in two families of flag codes with optimal properties and longer type vectors.

Findings

Constructed flag codes with optimal distance properties.

Achieved maximal cardinality for the new flag code families.

Unified cyclic orbit approach for longer type vectors.

Abstract

Flag codes generalize constant dimension codes by considering sequences of nested subspaces with prescribed dimensions as codewords. A comprehensive construction, which unites cyclic orbit flag codes, yields two families of flag codes on $\mathbb{F}^n_q$ (where $n=sk+h$ with $s\geq 2$ and $0\leq h < k$): optimum distance flag codes of the longest possible type vector $(1, 2, \ldots, k, n-k, \ldots, n-1)$ and flag codes with longer type vectors $(1, 2, \ldots, k+h, 2k+h, \ldots, (s-2)k+h, n-k, \ldots, n-1)$. These flag codes achieve the same cardinality $\sum^{s-1}_{i=1}q^{ik+h}+1$.