Cardinality-consistent flag codes with longer type vectors
Junfeng Jia, Yanxun Chang
TL;DR
This work addresses the problem of designing flag codes with longer type vectors while maintaining strong distance properties and large cardinality. It develops a unified construction based on cyclic orbits and companion matrices of primitive polynomials to produce two main families: optimum distance flag codes on $\mathbb{F}_q^{sk+h}$ with the longest type $\mathbf{t}=(1,2,\ldots,k,n-k,\ldots,n-1)$ and full flag codes on $\mathbb{F}_q^{2k+h}$, both achieving cardinality $\sum_{i=1}^{s-1}q^{ik+h}+1$; it also introduces flag codes on $\mathbb{F}_q^{sk+h}$ with longer type vectors $\mathbf{t}=(1,\ldots,k+h,2k+h,\ldots,(s-2)k+h,n-k,\ldots,n-1)$ that preserve the same cardinality and attain distance $d_f=2k(s+h+k-2)$ (except a special $s=4$ case). The results ensure that all projected codes reach maximal subspace distance, yielding cardinality-consistent flag codes with robust error-detection and correction capabilities. This systematic exploration advances flag code design for network coding by enabling longer information-bearing sequences without sacrificing cardinality or distance.
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 $\sum^{s-1}_{i=1}q^{ik+h}+1$.