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Vector Multispaces and Multispace Codes

Mladen Kovačević

TL;DR

It is shown that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes and extend them to an infinitely larger set of parameters.

Abstract

Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than $ 1 $ are derived. An application in coding theory is illustrated by showing that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes (defined in the set of all subspaces of $ \mathbb{F}_q^n $) and extend them to an infinitely larger set of parameters. In particular, in contrast to subspace codes, multispace codes of arbitrarily large cardinality and minimum distance exist for any fixed $ n $ and $ q $.

Vector Multispaces and Multispace Codes

TL;DR

It is shown that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes and extend them to an infinitely larger set of parameters.

Abstract

Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than are derived. An application in coding theory is illustrated by showing that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes (defined in the set of all subspaces of ) and extend them to an infinitely larger set of parameters. In particular, in contrast to subspace codes, multispace codes of arbitrarily large cardinality and minimum distance exist for any fixed and .
Paper Structure (3 sections, 7 theorems, 18 equations, 1 figure)

This paper contains 3 sections, 7 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Vector Multispaces
  3. Multispace Codes

Key Result

Proposition 2.3

A multiset $W$ over $\mathbb{F}_q^n$ is a multispace if and only if the following two conditions hold: (i) $\underline{W}$ is a subspace of $\mathbb{F}_q^n$, (ii) every element of $W$ has multiplicity $q^{t}$, for some nonnegative integer $t$. Furthermore, $t = \operatorname{rank}(W) - \dim(W)$.

Figures (1)

  • Figure 1: Hasse diagram of the set of all multispaces over $\mathbb{F}_2^3$ of rank up to $3$, partially ordered by inclusion. Subspaces of $\mathbb{F}_2^3$ are represented by light-blue nodes.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 5 more