Abstract
In network coding, a flag code is a set of sequences of nested subspaces of \({{\mathbb {F}}}_q^n\), being \({{\mathbb {F}}}_q\) the finite field with q elements. Flag codes defined as orbits of a cyclic subgroup of the general linear group acting on flags of \({{\mathbb {F}}}_q^n\) are called cyclic orbit flag codes. Inspired by the ideas in Gluesing-Luerssen et al. (Adv Math Commun 9(2):177–197, 2015), we determine the cardinality of a cyclic orbit flag code and provide bounds for its distance with the help of the largest subfield over which all the subspaces of a flag are vector spaces (the best friend of the flag). Special attention is paid to two specific families of cyclic orbit flag codes attaining the extreme possible values of the distance: Galois cyclic orbit flag codes and optimum distance cyclic orbit flag codes. We study in detail both classes of codes and analyze the parameters of the respective subcodes that still have a cyclic orbital structure.
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References
Ahlswede R., Cai N., Li R., Yeung R.W.: Network information flow. IEEE Trans. Inf. Theory 46, 1204–1216 (2000).
Alonso-González C., Navarro-Pérez M.A., Soler-Escrivà X.: An orbital construction of optimum distance flag codes. Finite Fields Appl. 73, 101861 (2021).
Alonso-González C., Navarro-Pérez M.A., Soler-Escrivà X.: Flag codes from planar spreads in network coding. Finite Fields Appl. 68, 101745 (2020).
Alonso-González C., Navarro-Pérez M.A., Soler-Escrivà X.: Optimum distance flag codes from spreads via perfect matchings in graphs. arXiv:2005.09370 (preprint).
Ben-Sasson E., Etzion T., Gabizon A., Raviv N.: Subspace polynomials and cyclic subspace codes. IEEE Trans. Inf. Theory 62, 1157–1165 (2016).
Chen B., Liu H.: Constructions of cyclic constant dimension codes. Des. Codes Cryptogr. 86(6), 1267–1279 (2018).
Drudge K.: On the orbits of singer groups and their subgroups. Electron. J. Comb. 9, R15 (2002).
Etzion T., Vardy A.: Error-correcting codes in projective space. IEEE Trans. Inf. Theory 57, 1165–1173 (2011).
Gluesing-Luerssen H., Lehmann H.: Distance distributions of cyclic orbit codes. Des. Codes Cryptogr. 89, 447–470 (2021).
Gluesing-Luerssen H., Morrison K., Troha C.: Cyclic orbit codes and stabilizer subfields. Adv. Math. Commun. 9(2), 177–197 (2015).
Gorla E., Manganiello F., Rosenthal J.: An algebraic approach for decoding spread codes. Adv. Math. Commun. 6(4), 443–466 (2012).
Ho T., Médard M., Koetter R., Karger D.R., Effros M., Shi J., Leong B.: A random linear network coding approach to multicast. IEEE Trans. Inf. Theory 52, 4413–4430 (2006).
Koetter R., Kschischang F.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54, 3579–3591 (2008).
Kurz S.: Bounds for flag codes. arXiv:2005.04768 (preprint).
Liebhold D., Nebe G., Vazquez-Castro A.: Network coding with flags. Des. Codes Cryptogr. 86(2), 269–284 (2018).
Manganiello F., Gorla E., Rosenthal J.: Spread codes and spread decoding in network coding. In: Proceedings of the 2008 IEEE International Symposium on Information Theory (ISIT), Toronto, Canada, pp. 851–855 (2008).
Manganiello F., Trautmann A.-L.: Spread decoding in extension fields. Finite Fields Appl. 25, 94–105 (2014).
Manganiello F., Trautmann A.-L., Rosenthal J.: On conjugacy classes of subgroups of the general linear group and cyclic orbit codes. In: Proceedings of the 2011 IEEE International Symposium on Information Theory (ISIT), Saint Pettersburg, pp. 1916–1920 (2011).
Nóbrega R.W., Uchôa-Filho B.F.: Multishot codes for network coding: bounds and a multilevel construction. In: 2009 IEEE International Symposium on Information Theory, Proceedings (ISIT), Seoul, South Korea, pp. 428–432 (2009).
Nóbrega R.W., Uchôa-Filho B.F.: Multishot codes for network coding using rank-metric codes. In: 2010 Third IEEE International Workshop on Wireless Network Coding, Boston, USA, pp. 1–6 (2010).
Otal K., Özbudak F.: Cyclic subspace codes via subspace polynomials. Des. Codes Cryptogr. 85(2), 191–204 (2017).
Rosenthal J., Trautmann A.-L.: A complete characterization of irreducible cyclic orbit codes and their Plücker embedding. Des. Codes Cryptogr. 66, 275–289 (2013).
Roth R.M., Raviv N., Tamo I.: Construction of sidon spaces with applications to coding. IEEE Trans. Inf. Theory 64(6), 4412–4422 (2018).
Segre B.: Teoria di Galois, Fibrazioni Proiettive e Geometrie non Desarguesiane. Annali di Matematica 64, 1–76 (1964).
Trautmann A.-L., Manganiello F., Braun M., Rosenthal J.: Cyclic orbit codes. IEEE Trans. Inf. Theory 59(11), 7386–7404 (2013).
Trautmann A.-L., Manganiello F., Rosenthal J.: Orbit codes: a new concept in the area of network coding. In: Proceedings of IEEE Information Theory Workshop, Dublin, Ireland, pp. 1–4 (2010).
Trautmann A.-L., Rosenthal J.: Constructions of constant dimension codes. In: Greferath M., et al. (eds.) Network Coding and Subspace Designs, pp. 25–42. E-Springer International Publishing AG (2018).
Zhao W., Tang X.: A characterization of cyclic subspace codes via subspace polynomials. Finite Fields Appl. 57, 1–12 (2019).
Acknowledgements
This study was supported by Ministerio de Ciencia e Innovación (Grant No. PID2019-108668GB-I00) and Conselleria d’Educació, Investigació, Cultura i Esport (Grant No. ACIF/2018/196).
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Alonso-González, C., Navarro-Pérez, M.Á. Cyclic orbit flag codes. Des. Codes Cryptogr. 89, 2331–2356 (2021). https://doi.org/10.1007/s10623-021-00920-5
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DOI: https://doi.org/10.1007/s10623-021-00920-5