Description
A code that is a set of subspaces of \(\mathbb{F}_q^n\). Codewords are generator matrices of the subspaces in reduced-row echelon form, and distance is governed by various notions of subspace overlap.Protection
Subspace codes are quantified with respect to the subspace distance [1] or injection distance [2].
Generalizations of various bounds for ordinary \(q\)-ary codes have been developed for subspace codes; see [4].
Decoding
List decoding up to the Singleton bound [5].Cousins
- Projective geometry code— Subspace codes are sets of subspaces of a projective space \(PG(n-1,q)\).
- Gabidulin code— Gabidulin codes can be used to construct asymptotically good subspace codes [1,8].
- Rank-metric code— An \(m \times n\) rank-metric codeword \(A\) can be lifted to a subspace codeword \((I | A)\) that generates an \(m\)-dimensional subspace [6][9; Def. 14.5.21].
- Poset code— Poset-code and subspace-code distance metric families intersect only at the Hamming metric [10].
Member of code lists
Primary Hierarchy
Parents
Subspace codes are represented by generator matrices of subspaces of \(\mathbb{F}_q^n\).
Subspace code
Children
Subspace codes of constant dimension reduce to constant-dimension codes.
References
- [1]
- R. Koetter and F. Kschischang, “Coding for Errors and Erasures in Random Network Coding”, (2008) arXiv:cs/0703061
- [2]
- D. Silva and F. R. Kschischang, “On Metrics for Error Correction in Network Coding”, IEEE Transactions on Information Theory 55, 5479 (2009) arXiv:0805.3824 DOI
- [3]
- M. Braun, T. Etzion, and A. Vardy, “Linearity and Complements in Projective Space”, (2011) arXiv:1103.3117
- [4]
- F. R. Kschischang, “Network Codes.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [5]
- V. Guruswami and C. Xing, “List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound”, Proceedings of the forty-fifth annual ACM symposium on Theory of Computing 843 (2013) DOI
- [6]
- D. Silva, F. R. Kschischang, and R. Koetter, “A Rank-Metric Approach to Error Control in Random Network Coding”, IEEE Transactions on Information Theory 54, 3951 (2008) arXiv:0711.0708 DOI
- [7]
- A. Khaleghi, D. Silva, and F. R. Kschischang, “Subspace Codes”, Lecture Notes in Computer Science 1 (2009) DOI
- [8]
- Huaxiong Wang, Chaoping Xing, and R. Safavi-Naini, “Linear authentication codes: bounds and constructions”, IEEE Transactions on Information Theory 49, 866 (2003) DOI
- [9]
- L. Storme, “Coding Theory and Galois Geometries.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [10]
- M. Firer, “Alternative Metrics.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
Page edit log
- Victor V. Albert (2024-08-23) — most recent
Cite as:
“Subspace code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/subspace