The Lattice Structure of Linear Subspace Codes
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The projective space P_q(n), i.e. the set of all subspaces of the vector space 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 F_2^n using a framework of lattices. They defined linear codes in P_q(n) by mimicking key features of linear codes in the Hamming space 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.
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