Subspace code

Subspace code[1]

A code that is a set of subspaces of \(GF(q)^n\).

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 [3].

List decoding up to the Singleton bound [4].

Packet-based transmission over networks [3].

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 [1,5].
Rank-metric code — Subspace and rank-metric codes are closely related [6].
Poset code — Poset-code and subspace-code distance metric families intersect only at the Hamming metric [7].

[1]: R. Koetter and F. R. Kschischang, “Coding for Errors and Erasures in Random Network Coding”, IEEE Transactions on Information Theory 54, 3579 (2008) DOI
[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]: F. R. Kschischang, "Network Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[4]: 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 (2013) DOI
[5]: Huaxiong Wang, Chaoping Xing, and R. Safavi-Naini, “Linear authentication codes: bounds and constructions”, IEEE Transactions on Information Theory 49, 866 (2003) 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) DOI
[7]: M. Firer, "Alternative Metrics." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI

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“Subspace code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/subspace