Construction of optimal flag codes by MRD codes

TL;DR

This paper introduces a new construction of ODFCs by maximum rank-metric codes, and proves that there is an (n,A)q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}

Abstract

Flag codes have received a lot of attention due to its application in random network coding. In 2021, Alonso-González et al. constructed optimal $(n,\mathcal{A})$-Optimum distance flag codes(ODFC) for $\mathcal {A}\subseteq \{1,2,\ldots,k,n-k,\ldots,n-1\}$ with $k\in \mathcal A$ and $k\mid n$. In this paper, we introduce a new construction of $(n,\mathcal A)_q$-ODFCs by maximum rank-metric codes, and prove that there is an $(n,\mathcal{A})$-ODFC of size $\frac{q^n-q^{k+r}}{q^k-1}+1$ for any $\mathcal{A}\subseteq\{1,2,\ldots,k,n-k,\ldots,n-1\}$ with $\mathcal A\cap \{k,n-k\}\neq\emptyset$, where $r\equiv n\pmod k$ and $0\leq r<k$. Furthermore, when $k>\frac{q^r-1}{q-1}$, this $(n,\mathcal A)_q$-ODFC is optimal. Specially, when $r=0$, Alonso-González et al.'s result is also obtained. We also gives a characterization of almost optimum distance flag codes, and construct a family of optimal almost optimum flag distance codes.

Theorems & Definitions (25)

• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Theorem 2.8
• Remark 2.9
• Theorem 2.10