Abstract
Orbit codes, as special constant dimension codes, have attracted much attention due to their applications for error correction in random network coding. They arise as orbits of a subspace of \({\mathbb {F}}^n_q\) under the action of some subgroup of the finite general linear group \(\textrm{GL}_n(q)\). The aim of this paper is to present constructions of large non-Abelian orbit codes having the maximum possible distance. The properties of imprimitive wreath products and wreathed tensor products of groups are employed to select certain types of subspaces and their stabilizers, thereby providing a systematic way of constructing orbit codes with optimum parameters. We also present explicit examples of such constructions which improve the parameters of the construction already obtained in Climent et al. (Cryptogr Commun 11:839-852, 2019).
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Ahlswede, R., Cai, N., Li, S.-Y.R., Yeung, R.W.: Network information flow. IEEE Trans. Inf. Theory 46(4), 1204–1216 (2000)
Bardestani, F., Iranmanesh, A.: Cyclic orbit codes with the normalizer of a Singer subgroup. J. Sci. Islamic Republic Iran 26(1), 49–55 (2015)
Bastos, G., Junior, R.P., Guerreiro, M.: Abelian noncyclic orbit codes and multishot subspace codes. Adv. Math. Commun. 14, 631–650 (2020)
Bray, J.N., Holt, D., Roney-Dougal, C.: The Maximal Subgroups of the Low-Dimensional Finite Classical Groups. Cambridge University Press, Cambridge (2013)
Ben-Sasson, E., Etzion, T., Gabizon, A., Raviv, N.: Subspace polynomials and cyclic subspace codes. IEEE Trans. Inf. Theory 62(3), 1157–1165 (2016)
Chen, S.D., Liang, J.Y.: New Constructions of Orbit Codes Based on the Operations of Orbit Codes. Acta Math. Appl. Sin. Engl. Ser. 36, 803–815 (2020)
Climent, J.J., Requena, V., Soler-Escriva, X.: A construction of Abelian non-cyclic orbit codes. Cryptogr. Commun. 11, 839–852 (2019)
Coutts, H.J., Quick, M., Roney-Dougal, C.M.: The primitive permutation groups of degree less than 4096. Commun. Algebra 39(10), 3526–3546 (2011)
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A.: ATLAS of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press, Eynsham, (1985)
Feng, T., Wang, Y.: New constructions of large cyclic subspace codes and Sidon spaces. Discrete Math. 344(4), 7 (2021)
Gluesing-Luerssen, H., Lehmann, H.: Distance distributions of cyclic orbit codes. Des. Codes Cryptography 89(3), 447–470 (2021)
Gluesing-Luerssen, H., Troha, C.: Construction of subspace codes through linkage. Adv. Math. Commun. 10, 525–540 (2017)
Gluesing-Luerssen, H., Morrison, K., Troha, C.: Cyclic orbit codes and stabilizer subfields. Adv. Math. Commun. 9(2), 177–197 (2015)
Kotter, R., Kschischang, F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54(8), 3579–3591 (2008)
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 2008), pp. 881-885, Toronto, Canada. IEEE (2008)
OEIS: Number of primitive permutation groups of degree \(n\), In: The on-line encyclopedia of integer sequences (OEIS). Available: https://oeis.org/A000019
Rose, H. E.: A course on finite groups. Universitext. Springer-Verlag London, Ltd., London, xii+311 pp, (2009)
Rosenthal, J., Trautmann, A.-L.: A complete characterization of irreducible cyclic orbit codes and their Plucker embedding. Des. Codes Crypt. 66, 275–289 (2013)
Schneider, H. J.: Hopfalgebren und Quantengruppen, lecture notes at Ludwig-Maximilians-Universität München taken by D. Grinberg (German). Available: http://www.cip.ifi.lmu.de/~grinberg/algebra/hopf.pdf
Trautmann, A.L., Manganiello, F., Rosenthal, J.: Orbit codes—a new concept in the area of network coding. In: Proceedings of the 2010 IEEE information theory workshop (ITW 2010), Dublin, Ireland. IEEE (2010)
Wilson, R.A.: The Finite Simple Groups, Graduate Text in Mathematics, vol. 251. Springer-Verlag, London (2009)
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The authors would like to thank the anonymous referee for careful reading of the manuscript and providing invaluable and precise comments which improved the exposition of the paper significantly.
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Askary, S., Biranvand, N. & Shirjian, F. New constructions of orbit codes based on imprimitive wreath products and wreathed tensor products. Rend. Circ. Mat. Palermo, II. Ser 73, 85–98 (2024). https://doi.org/10.1007/s12215-023-00903-6
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DOI: https://doi.org/10.1007/s12215-023-00903-6