Grassmannian code[1–3] 

Description

Evaluation code of polynomials evaluated on points lying on a Grassmannian \({\mathbb{G}}(\ell,m)\).

Parent

  • Flag-variety code — Grassmannian codes are flag-variety evaluation codes with the flag variety being a Grassmannian.

Cousins

  • Griesmer code — The binary Grassmannian \([35,6,16]\) code, whose points lie on the Grassmannian \({\mathbb{G}(2,4)}\), attains the Griesmer bound [4].
  • Schubert code — Schubert varieties are subvarieties of Grassmannians, and Schubert codes were initially constructed as a generalization of Grassmannian codes.

References

[1]
C. T. Ryan, An application of Grassmannian varieties to coding theory. Congr. Numer. 57 (1987) 257–271.
[2]
C.T. Ryan, Projective codes based on Grassmann varieties, Congr. Numer. 57, 273–279 (1987).
[3]
C. T. Ryan and K. M. Ryan, “The minimum weight of the Grassmann codes C(k,n),”, Discrete Applied Mathematics 28, 149 (1990) DOI
[4]
J. B. Little, “Algebraic geometry codes from higher dimensional varieties”, (2008) arXiv:0802.2349
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Zoo Code ID: grassmannian

Cite as:
“Grassmannian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/grassmannian
BibTeX:
@incollection{eczoo_grassmannian,
  title={Grassmannian code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/grassmannian}
}
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Cite as:

“Grassmannian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/grassmannian

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/ag/varieties/grassmannian.yml.