New Constructions of Full Flag Codes Based on Partial Spreads

TL;DR

This paper introduces new methods for constructing full flag codes using partial spreads, achieving various optimality levels and providing an efficient decoding algorithm for improved network coding performance.

Contribution

It presents a novel family of full flag code constructions based on partial spreads, including codes with maximum and second-maximum distances, and introduces an efficient decoding algorithm.

Findings

Constructed flag codes with maximum and quasi-maximum distances.

Developed an efficient decoding algorithm for the proposed codes.

Resembles a 'sandwich' structure with layered partial spreads.

Abstract

Flag codes are a class of multishot network codes comprising sequences of nested subspaces (flags) within the vector space $\mathbb{F}_q^n$, where $q$ is a prime power. In this paper, we propose a family of constructions for full flag codes based on partial spreads. The distances of this family include maximum distance (optimum distance flag codes), second-maximum distance (quasi-optimum distance flag codes), as well as other feasible values. The structure of these flag codes resembles that of a \textquotedblleft sandwich", consisting of one layer of companion matrix and two layers of partial spreads. Furthermore, we present an efficient decoding algorithm for these codes.