Consistent Flag Codes

TL;DR

This paper investigates consistent flag codes over finite fields, analyzing their parameters, properties, and decoding algorithms, with a focus on their relation to projected codes and generalizations of equidistant and sunflower codes.

Contribution

It introduces the concept of consistent flag codes, explores their properties, and develops a decoding algorithm leveraging their unique consistency condition.

Findings

Consistent flag codes' parameters are determined by their projected codes.

Generalizations of equidistant and sunflower codes are presented for flag codes.

A decoding algorithm exploiting the consistency condition is proposed.

Abstract

In this paper we study flag codes on $\mathbb{F}_q^n$, being $\mathbb{F}_q$ the finite field with $q$ elements. Special attention is given to the connection between the parameters and properties of a flag code and the ones of a family of constant dimension codes naturally associated to it (the projected codes). More precisely, we focus on consistent flag codes, that is, flag codes whose distance and size are completely determined by their projected codes. We explore some aspects of this family of codes and present examples of them by generalizing the concepts of equidistant and sunflower subspace code to the flag codes setting. Finally, we present a decoding algorithm for consistent flag codes that fully exploits the consistency condition.