Quasi-optimum distance flag codes

TL;DR

This paper investigates quasi-optimum distance flag codes in network coding, characterizing their properties, establishing bounds, and proposing systematic constructions using partial spreads and sunflowers.

Contribution

It introduces a systematic construction method for quasi-optimum distance flag codes and extends the approach to lower minimum distances.

Findings

Characterization of quasi-optimum distance flag codes.

Upper bounds for the cardinality of such codes.

Construction methods using partial spreads and sunflowers.

Abstract

A flag is a sequence of nested subspaces of a given ambient space F_q^n over a finite field F_q. In network coding, a flag code is a set of flags, all of them with the same sequence of dimensions, the type vector. In this paper, we investigate quasi-optimum distance flag codes, i.e., those attaining the second best possible distance value. We characterize them and present upper bounds for their cardinality. Moreover, we propose a systematic construction for every choice of the type vector by using partial spreads and sunflowers. For flag codes with lower minimum distance, we adapt the previous construction and provide some results towards their characterization, especially in the case of the third best possible distance value.