Description
Evaluation code of polynomials evaluated on points lying on a finite-field Grassmannian \({\mathbb{G}}(\ell,m)\) embedded into projective space using the Plucker embedding [4,5].Cousins
- Griesmer code— The \([35,6,16]\) Grassmannian evaluation code, whose points lie on the Grassmannian \({\mathbb{G}(2,4)}\), attains the Griesmer bound [6].
- Finite-field Grassmannian code— Grassmannian evaluation codes are evaluation codes of polynomials evaluated on points lying on a finite-field Grassmannian \({\mathbb{G}}(\ell,m)\) embedded into projective space using the Plucker embedding [4,5].
- Schubert code— Schubert varieties are subvarieties of Grassmannians, and Schubert codes were initially constructed as a generalization of Grassmannian evaluation codes.
Member of code lists
Primary Hierarchy
Parents
Grassmannian evaluation codes are flag-variety evaluation codes with the flag variety being a Grassmannian.
Grassmannian evaluation code
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]
- D. Yu. Nogin, “Codes associated to Grassmannians”, Arithmetic, Geometry, and Coding Theory DOI
- [5]
- V. Lakshmibai and J. Brown, The Grassmannian Variety (Springer New York, 2015) DOI
- [6]
- J. B. Little, “Algebraic geometry codes from higher dimensional varieties”, (2008) arXiv:0802.2349
Page edit log
- Victor V. Albert (2022-08-10) — most recent
Cite as:
“Grassmannian evaluation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/grassmannian_variety