Linear subspace code [1]
Description
Subspace code equipped with an addition \(\boxplus\) under which it is an elementary Abelian 2-group with the zero subspace as identity and the subspace distance is translation invariant [1]. Such codes are projective-space analogues of linear binary codes, and their size is a power of two [1]. The dimension of \(X \boxplus Y\) equals the subspace distance between codewords \(X\) and \(Y\) [1].
The spans of all subsets of a fixed basis of \(\mathbb{F}_2^n\), with the symmetric difference of index sets as addition, form the code derived from a fixed basis [1].
Protection
A linear subspace code in \(\mathbb{F}_2^n\) has at most \(2^n\) codewords [2]. The bound is attained by the code derived from a fixed basis and by the Fano-plane code, whose nonzero codewords are the seven two-dimensional subspaces of \(\mathbb{F}_2^3\) [1,3]. If the full space \(\mathbb{F}_2^n\) is a codeword, a code attaining the bound is derived from a fixed basis [4]. For every prime power \(q>2\), some \(\mathbb{F}_q^n\) admits a linear subspace code with more than \(2^n\) codewords [3].Cousin
- Linear binary code— Linear subspace codes are projective-space analogues of linear binary codes, with the subspace distance and the addition \(\boxplus\) playing the roles of the Hamming distance and vector addition [1].
Member of code lists
Primary Hierarchy
References
- [1]
M. Braun, T. Etzion, and A. Vardy, “Linearity and Complements in Projective Space”, (2011) arXiv:1103.3117
- [2]
S. B. Pai, “A proof of the Braun-Etzion-Vardy bound for binary subspace codes”, (2026) arXiv:2609.33719
- [3]
P. Basu, “Equidistant Linear Codes in Projective Spaces”, (2021) arXiv:2107.10820
- [4]
S. Pai and B. S. Rajan, “On the Bounds of Certain Maximal Linear Codes in a Projective Space”, (2014) arXiv:1410.2725
Page edit log
- Victor V. Albert (2026-10-16) — most recent
Cite as:
“Linear subspace code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/linear_subspace, arXiv:2606.11484