LCPs of Subspace Codes
Sanjit Bhowmick
TL;DR
This work extends the theory of linear complementary pairs to subspace codes, introducing a framework where an LCP $\{\mathcal{C},\mathcal{D}\}$ satisfies $\mathcal{C}\oplus\mathcal{D}=\mathbb{F}_q^n$ and $\mathcal{C}\cap\mathcal{D}=\{0\}$, with a focus on the subspace distance $d_s$. It provides necessary and sufficient conditions for LCP existence via distance criteria and duality, and offers practical matrix-based criteria using generator and parity-check matrices. The paper then develops several explicit constructions of LCPs, including $[u|u+v]$-type (Plotkin) constructions, $[u+v|\lambda u-\lambda v]$-constructions, and $k$-spread based partitions, showing how LCPs are preserved under these operations. An application to insertion error correction demonstrates the practical utility of LCPs in network coding, with detection and correction strategies leveraging the complementary structure. Overall, the results advance the design of error- and attack-resilient subspace codes for secure and reliable communications in networked settings, providing a versatile toolkit for constructing and applying LCPs.
Abstract
A subspace code is a nonempty collection of subspaces of the vector space $\mathbb{F}_q^{n}$. A pair of linear codes is called a linear complementary pair (in short LCP) of codes if their intersection is trivial and the sum of their dimensions equals the dimension of the ambient space. Equivalently, the two codes form an LCP if the direct sum of these two codes is equal to the entire space. In this paper, we introduce the concept of LCPs of subspace codes. We first provide a characterization of subspace codes that form an LCP. Furthermore, we present a sufficient condition for the existence of an LCP of subspace codes based on a complement function on a subspace code. In addition, we give several constructions of LCPs for subspace codes using various techniques and provide an application to insertion error correction.