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Quasi-optimum distance flag codes

Clementa Alonso-González, Miguel Ángel Navarro-Pérez

TL;DR

This work investigates flag codes in network coding that achieve the second-best flag distance, called quasi-optimum distance flag codes (QODFC). It characterizes QODFCs by relating them to a central distinguished projected flag code of length at most four and derives sharp cardinality bounds via spreads, partial spreads, and sunflowers; it also provides systematic constructions for every type vector, including both disjoint and non-disjoint cases, and extends the analysis to the third-best distance value. The authors further examine lower-distance regimes, showing how distance losses distribute across projected codes and establishing a framework for building flag codes with distances $D^{(\mathbf{t}, n)}-2\ell$ using nested sunflowers. Duality is employed to reduce the problem to dual types, enabling broad applicability of the constructions. Overall, the paper advances near-optimal flag-code design, offering practical constructions and bounds across general type vectors for network coding applications.

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.

Quasi-optimum distance flag codes

TL;DR

This work investigates flag codes in network coding that achieve the second-best flag distance, called quasi-optimum distance flag codes (QODFC). It characterizes QODFCs by relating them to a central distinguished projected flag code of length at most four and derives sharp cardinality bounds via spreads, partial spreads, and sunflowers; it also provides systematic constructions for every type vector, including both disjoint and non-disjoint cases, and extends the analysis to the third-best distance value. The authors further examine lower-distance regimes, showing how distance losses distribute across projected codes and establishing a framework for building flag codes with distances using nested sunflowers. Duality is employed to reduce the problem to dual types, enabling broad applicability of the constructions. Overall, the paper advances near-optimal flag-code design, offering practical constructions and bounds across general type vectors for network coding applications.

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.
Paper Structure (12 sections, 25 theorems, 55 equations, 2 tables)

This paper contains 12 sections, 25 theorems, 55 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subspace codes
  4. Flag codes
  5. Quasi-optimum distance flag codes
  6. Characterization for QODFCs
  7. Bounds for QODFC
  8. Constructions based on spreads and sunflowers
  9. The non-disjoint case
  10. The disjoint case
  11. A study for lower distances: a systematic construction
  12. Conclusions and future work

Key Result

Proposition 2.1

Let $\mathcal{C}\subseteq\mathcal{G}_q(k,n)$ with $k\leqslant \frac{n}{2}.$ The code $\mathcal{C}$ has minimum distance $d_S(\mathcal{C})=2k$ if, and only if, every pair of different subspaces $\mathcal{U}, \mathcal{V}\in\mathcal{C}$ intersects trivially.

Theorems & Definitions (66)

  • Proposition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 56 more