On Linear Codes in Projective Spaces

Abstract

The projective space $\mathbb{P}_q(n)$ of order $n$ over a finite field $\mathbb{F}_q$ is defined as the collection of all subspaces of the ambient space $\mathbb{F}_q^n$. The Grassmannian $\mathcal{G}_q(n, k)$ is the set of all members of $\mathbb{P}_q(n)$ with fixed dimension $k$. The subspace distance function, defined as $d_S(X, Y) = \dim(X+Y) - \dim(X \cap Y)$ serves as a suitable metric to turn the projective space $\mathbb{P}_q(n)$ into a metric space. A code in the projective space $\mathbb{P}_q(n)$ is a subset of $\mathbb{P}_q(n)$. Projective space has been shown previously by Koetter and Kschischang to be the ideal coding space for error and erasure correction in random network coding. Linear codes find huge applications in classical error-correction. The notion of linearity was introduced in codes in projective space recently. A subspace code $\mathcal{U}$ in $\mathbb{P}_q(n)$ that contains $\left\{ 0\right\}$ is linear if there exists a function $\boxplus : \mathcal{U} \times \mathcal{U} \rightarrow \mathcal{U}$ such that $(\mathcal{U}, \boxplus)$ is an abelian group with identity element as $\left\{ 0\right\}$, all elements of $\mathcal{U}$ are idempotent with respect to $\boxplus$, and the operation $\boxplus$ is isometric. It was conjectured that the size of any linear subspace code in $\mathbb{P}_q(n)$ can be at most $2^n$. In this work, we focus on different classes of linear subspace codes with a view to proving the conjectured upper bound for them as well as characterizing the maximal cases. We study connections of linear codes with lattices and a few combinatorial objects. Binary linear block codes and linear subspace codes are subspaces of a finite vector space over $\mathbb{F}_2$. We identify common features in their structures and prove analogous results for subspace codes including the Union-Intersection theorem. We investigate a class of linear subspace codes which are closed under intersection and show that these codes are equivalent to codes derived from a partition of a linearly ind...