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doi: 10.3934/amc.2021027
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Degenerate flag varieties in network coding

Ghislain Fourier 1, and  Gabriele Nebe 1,,

RWTH Aachen University, 52056 Aachen, Germany

* Corresponding author: Gabriele Nebe

Received  March 2021 Early access July 2021

Fund Project: This is a contribution to Project-ID 286237555 - TRR 195 - by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

Building upon the application of flags to network coding introduced in [6], we develop a variant of this coding technique that uses degenerate flags. The information set is a metric affine space isometric to the space of upper triangular matrices endowed with the flag rank metric. This suggests the development of a theory for flag rank metric codes in analogy to the rank metric codes used in linear subspace coding.

Keywords: Network coding, degenerate flag variety, flag rank metric codes, upper triangular matrices, error correcting codes, Grassmann distance on flags.
Mathematics Subject Classification: Primary: 20E42, 94B99; Secondary: 20B30.
Citation: Ghislain Fourier, Gabriele Nebe. Degenerate flag varieties in network coding. Advances in Mathematics of Communications, doi: 10.3934/amc.2021027
References:
[1]

G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier and R. Reineke, Linear degenerations of flag varieties, Math. Z., 287 (2017), 615-654.  doi: 10.1007/s00209-016-1839-y.  Google Scholar

[2]

E. Feigin, $\mathbb{G}_a^M$ degeneration of flag varieties, Selecta Math. (N.S.), 18 (2012), 513-537.  doi: 10.1007/s00029-011-0084-9.  Google Scholar

[3]

E. Feigin, G. Fourier and P. Littelmann, PBW filtration and bases for irreducible modules in type $ {\mathbb{A}} _n$, Transform. Groups, 16 (2011), 71-89.  doi: 10.1007/s00031-010-9115-4.  Google Scholar

[4]

G. C. Irelli and M. Lanini, Degenerate flag varieties of type $ {\mathbb{A}} $ and $ {\mathbb{C}} $ are schubert varieties, Int. Math. Research Notices, 2015 (2015), 6353-6374.  doi: 10.1093/imrn/rnu128.  Google Scholar

[5]

R. Kötter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform., 54 (2008), 3579-3591.  doi: 10.1109/TIT.2008.926449.  Google Scholar

[6]

D. Liebhold, G. Nebe and A. Vazquez-Castro, Network coding with flags, Des. Codes Cryptogr., 86 (2018), 269-284.  doi: 10.1007/s10623-017-0361-5.  Google Scholar

[7]

D. Liebhold, Flag Codes With Application to Network Coding, PhD Thesis, RWTH Aachen 2019. Google Scholar

[8]

A. Ravagnani, Rank-metric codes and their duality theory, Des. Codes Cryptogr., 80 (2016), 197-216.  doi: 10.1007/s10623-015-0077-3.  Google Scholar

[9]

D. Silva, F. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.  doi: 10.1109/TIT.2008.928291.  Google Scholar

show all references

References:
[1]

G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier and R. Reineke, Linear degenerations of flag varieties, Math. Z., 287 (2017), 615-654.  doi: 10.1007/s00209-016-1839-y.  Google Scholar

[2]

E. Feigin, $\mathbb{G}_a^M$ degeneration of flag varieties, Selecta Math. (N.S.), 18 (2012), 513-537.  doi: 10.1007/s00029-011-0084-9.  Google Scholar

[3]

E. Feigin, G. Fourier and P. Littelmann, PBW filtration and bases for irreducible modules in type $ {\mathbb{A}} _n$, Transform. Groups, 16 (2011), 71-89.  doi: 10.1007/s00031-010-9115-4.  Google Scholar

[4]

G. C. Irelli and M. Lanini, Degenerate flag varieties of type $ {\mathbb{A}} $ and $ {\mathbb{C}} $ are schubert varieties, Int. Math. Research Notices, 2015 (2015), 6353-6374.  doi: 10.1093/imrn/rnu128.  Google Scholar

[5]

R. Kötter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform., 54 (2008), 3579-3591.  doi: 10.1109/TIT.2008.926449.  Google Scholar

[6]

D. Liebhold, G. Nebe and A. Vazquez-Castro, Network coding with flags, Des. Codes Cryptogr., 86 (2018), 269-284.  doi: 10.1007/s10623-017-0361-5.  Google Scholar

[7]

D. Liebhold, Flag Codes With Application to Network Coding, PhD Thesis, RWTH Aachen 2019. Google Scholar

[8]

A. Ravagnani, Rank-metric codes and their duality theory, Des. Codes Cryptogr., 80 (2016), 197-216.  doi: 10.1007/s10623-015-0077-3.  Google Scholar

[9]

D. Silva, F. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.  doi: 10.1109/TIT.2008.928291.  Google Scholar

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