The Lattice Structure of Linear Subspace Codes

The projective space ${\mathbb{P}}_q(n)$, i.e. the set of all subspaces of the vector space ${\mathbb{F}}_q^n$, is a metric space endowed with the subspace distance metric. Braun, Etzion and Vardy argued that codes in a projective space are analogous to binary block codes in ${\mathbb{F}}_2^n$ using a framework of lattices. They defined linear codes in ${\mathbb{P}}_q(n)$ by mimicking key features of linear codes in the Hamming space ${\mathbb{F}}_2^n$. In this paper, we prove that a linear code in a projective space forms a sublattice of the corresponding projective lattice if and only if the code is closed under intersection. The sublattice thus formed is geometric distributive. We also present an application of this lattice-theoretic characterization.