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.

翻译：子空间码是向量空间 $\mathbb{F}_q^{n}$ 中非空子空间的集合。若两个线性码的交集是平凡的且它们的维数之和等于环境空间的维数，则称这对线性码为线性互补对（简称LCP）。等价地，若这两个码的直和等于整个空间，则它们构成一个LCP。本文引入了子空间码的线性互补对概念。我们首先刻画了构成LCP的子空间码的特征。此外，基于子空间码上的补函数，我们提出了子空间码LCP存在的一个充分条件。另外，我们利用多种技术给出了子空间码LCP的若干构造，并提供了在插入错误纠正中的一个应用。