Abstract

This paper defines the $q$-analogue of a matroid and establishes several properties like duality, restriction and contraction. We discuss possible ways to define a $q$-matroid, and why they are (not) cryptomorphic. Also, we explain the motivation for studying $q$-matroids by showing that a rank metric code gives a $q$-matroid.

Highlights

  • This paper establishes the definition and several basic properties of q-matroids

  • We explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid

  • We show that every rank metric code gives rise to a q-matroid

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Summary

Introduction

This paper establishes the definition and several basic properties of q-matroids. we explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid. The “q” in q-analogue does refer to quantum, and to the size of a finite field In the latter case, [n]q is the number of 1-dimensional subspaces of a vector space Fnq ; and in general, we can view 1-dimensional subspaces of a finite dimensional space as the q-analogues of the elements of a finite set. If we consider q as the size of a finite field, the q-binomial counts the number of subspaces of Fnq of dimension k. Using the quotient space as a complement will lower the dimension of the ambient space, which makes it perfect for the definition of contraction but not very suitable for other purposes The solution to this problem is to use all options described above, depending on for which property of A − B we need a q-analogue

Rank function
Independent spaces
Rank metric codes
Truncation
Duality
Restriction and contraction
10 Towards more cryptomorphisms
10.1 Circuits
10.2 Closure
11 Further research directions

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