{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,12]],"date-time":"2024-08-12T11:40:50Z","timestamp":1723462850197},"reference-count":50,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["AMC"],"published-print":{"date-parts":[[2022]]},"abstract":"<p style='text-indent:20px;'>In this paper we construct different families of orbit codes in the vector spaces of the symmetric bilinear forms, quadratic forms and Hermitian forms on an <inline-formula><tex-math id=\"M1\">\\begin{document}$ n $\\end{document}<\/tex-math><\/inline-formula>-dimensional vector space over the finite field <inline-formula><tex-math id=\"M2\">\\begin{document}$ {\\mathbb F_{q}} $\\end{document}<\/tex-math><\/inline-formula>. All these codes admit the general linear group <inline-formula><tex-math id=\"M3\">\\begin{document}$ {{{{\\rm{GL}}}}}(n,q) $\\end{document}<\/tex-math><\/inline-formula> as a transitive automorphism group.<\/p><\/jats:p>","DOI":"10.3934\/amc.2020105","type":"journal-article","created":{"date-parts":[[2020,8,24]],"date-time":"2020-08-24T08:47:40Z","timestamp":1598258860000},"page":"135","source":"Crossref","is-referenced-by-count":1,"title":["Orbit codes from forms on vector spaces over a finite field"],"prefix":"10.3934","volume":"16","author":[{"given":"Angela","family":"Aguglia","sequence":"first","affiliation":[]},{"given":"Antonio","family":"Cossidente","sequence":"additional","affiliation":[]},{"given":"Giuseppe","family":"Marino","sequence":"additional","affiliation":[]},{"given":"Francesco","family":"Pavese","sequence":"additional","affiliation":[]},{"given":"Alessandro","family":"Siciliano","sequence":"additional","affiliation":[]}],"member":"2321","reference":[{"key":"key-10.3934\/amc.2020105-1","doi-asserted-by":"publisher","unstructured":"R. Ahlswede, N. Cai, S.-Y. Li, R. W. Yeung.Network information flow, IEEE Trans. Inform. Theory<\/i>, 46<\/b> (2000), 1204-1216.","DOI":"10.1109\/18.850663"},{"key":"key-10.3934\/amc.2020105-2","unstructured":"M. Aschbacher., Finite Group Theory<\/i>, ${ref.volume}<\/b> (1986)."},{"key":"key-10.3934\/amc.2020105-3","doi-asserted-by":"publisher","unstructured":"E. Ben-Sasson, T. Etzion, A. Gabizon, N. Raviv.Subspace polynomials and cyclic subspace codes, IEEE Trans. Inform. Theory<\/i>, 62<\/b> (2016), 1157-1165.","DOI":"10.1109\/TIT.2016.2520479"},{"key":"key-10.3934\/amc.2020105-4","unstructured":"O. Bottema.On the Betti-Mathieu group, Nieuw Arch. Wisk.<\/i>, 16<\/b> (1930), 46-50."},{"key":"key-10.3934\/amc.2020105-5","doi-asserted-by":"publisher","unstructured":"M. Braun, T. Etzion, P. R. J. \u00d6sterg\u00e5rd, A. Vardy and A. Wassermann, Existence of $q$\u2013analogs of Steiner systems, Forum Math. Pi<\/i>, 4<\/b> (2016), e7, 14 pp.","DOI":"10.1017\/fmp.2016.5"},{"key":"key-10.3934\/amc.2020105-6","unstructured":"L. Carlitz.A Note on the Betti-Mathieu group, Portugaliae mathematica<\/i>, 22<\/b> (1963), 121-125."},{"key":"key-10.3934\/amc.2020105-7","doi-asserted-by":"publisher","unstructured":"B. Chen, H. Liu.Constructions of cyclic constant dimension codes, Des. Codes Cryptogr.<\/i>, 86<\/b> (2018), 1267-1279.","DOI":"10.1007\/s10623-017-0394-9"},{"key":"key-10.3934\/amc.2020105-8","doi-asserted-by":"publisher","unstructured":"J.-J. Climent, V. Requena, X. Soler-Escriv\u00e0.A construction of Abelian non-cyclic orbit codes, Cryptography and Communication<\/i>, 11<\/b> (2019), 839-852.","DOI":"10.1007\/s12095-018-0306-5"},{"key":"key-10.3934\/amc.2020105-9","doi-asserted-by":"crossref","unstructured":"B. N. Cooperstein.External flats to varieties in ${{{\\rm PG}}}(M_{n, n}({{{\\rm GF}}}(q)))$, Linear Algebra Appl.<\/i>, 267<\/b> (1997), 175-186.","DOI":"10.1016\/S0024-3795(97)80049-3"},{"key":"key-10.3934\/amc.2020105-10","doi-asserted-by":"publisher","unstructured":"A. Cossidente, F. Pavese.On subspace codes, Des. Codes Cryptogr.<\/i>, 78<\/b> (2016), 527-531.","DOI":"10.1007\/s10623-014-0018-6"},{"key":"key-10.3934\/amc.2020105-11","doi-asserted-by":"publisher","unstructured":"A. Cossidente, F. Pavese.Veronese subspace codes, Des. Codes Cryptogr.<\/i>, 81<\/b> (2016), 445-457.","DOI":"10.1007\/s10623-015-0166-3"},{"key":"key-10.3934\/amc.2020105-12","doi-asserted-by":"publisher","unstructured":"A. Cossidente, F. Pavese.Subspace codes in ${{{\\rm PG}}}(2n-1, q)$, Combinatorica<\/i>, 37<\/b> (2017), 1073-1095.","DOI":"10.1007\/s00493-016-3354-5"},{"key":"key-10.3934\/amc.2020105-13","doi-asserted-by":"crossref","unstructured":"A. Cossidente, F. Pavese and L. Storme, Geometrical aspects of subspace codes, in Network Coding and Subspace Designs<\/i>, 107\u2013129, Signals Commun. Technol., Springer, Cham, 2018.","DOI":"10.1007\/978-3-319-70293-3_6"},{"key":"key-10.3934\/amc.2020105-14","doi-asserted-by":"publisher","unstructured":"A. Cossidente, F. Pavese, L. Storme.Optimal subspace codes in ${{{\\rm PG}}}(4, q)$, Adv. Math. Commun.<\/i>, 13<\/b> (2019), 393-404.","DOI":"10.3934\/amc.2019025"},{"key":"key-10.3934\/amc.2020105-15","unstructured":"A. Cossidente, S. Kurz, G. Marino and F. Pavese, Combining subspace codes, preprint, arXiv: 1911.03387."},{"key":"key-10.3934\/amc.2020105-16","doi-asserted-by":"publisher","unstructured":"B. Csajb\u00f3k, A. Siciliano.Puncturing maximum rank distance codes, J. Algebraic Combin.<\/i>, 49<\/b> (2019), 507-534.","DOI":"10.1007\/s10801-018-0833-3"},{"key":"key-10.3934\/amc.2020105-17","doi-asserted-by":"crossref","unstructured":"P. Dembowski, Finite Geometries<\/i>, Springer-Verlag, Berlin-New York, 1968.","DOI":"10.1007\/978-3-642-62012-6"},{"key":"key-10.3934\/amc.2020105-18","doi-asserted-by":"crossref","unstructured":"N. Durante and A. Siciliano, Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries, Electron. J. Combin.<\/i>, 24<\/b> (2017), 18 pp.","DOI":"10.37236\/6106"},{"key":"key-10.3934\/amc.2020105-19","doi-asserted-by":"publisher","unstructured":"T. Etzion, N. Silberstein.Error-correcting codes in projective spaces via rank- metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory<\/i>, 55<\/b> (2009), 2909-2919.","DOI":"10.1109\/TIT.2009.2021376"},{"key":"key-10.3934\/amc.2020105-20","doi-asserted-by":"publisher","unstructured":"T. Etzion, N. Silberstein.Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory<\/i>, 59<\/b> (2013), 1004-1017.","DOI":"10.1109\/TIT.2012.2220119"},{"key":"key-10.3934\/amc.2020105-21","doi-asserted-by":"publisher","unstructured":"T. Etzion, A. Vardy.Error-correcting codes in projective space, IEEE Trans. Inform. Theory<\/i>, 57<\/b> (2011), 1165-1173.","DOI":"10.1109\/TIT.2010.2095232"},{"key":"key-10.3934\/amc.2020105-22","doi-asserted-by":"publisher","unstructured":"G. Faina, G. Kiss, S. Marcugini, F. Pambianco.The cyclic model for ${{{\\rm PG}}}(n-1, q)$ and a construction of arcs, European J. Combin.<\/i>, 23<\/b> (2002), 31-35.","DOI":"10.1006\/eujc.2001.0525"},{"key":"key-10.3934\/amc.2020105-23","doi-asserted-by":"publisher","unstructured":"H. Gluesing-Luerssen, K. Morrison, C. Troha.Cyclic orbit codes and stabilizer subfields, Adv. Math. Commun.<\/i>, 9<\/b> (2015), 177-197.","DOI":"10.3934\/amc.2015.9.177"},{"key":"key-10.3934\/amc.2020105-24","doi-asserted-by":"publisher","unstructured":"H. Gluesing-Luerssen, C. Troha.Construction of subspace codes through linkage, Adv. Math. Commun.<\/i>, 10<\/b> (2016), 525-540.","DOI":"10.3934\/amc.2016023"},{"key":"key-10.3934\/amc.2020105-25","unstructured":"D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864, 2016."},{"key":"key-10.3934\/amc.2020105-26","doi-asserted-by":"crossref","unstructured":"J. W. P. Hirschfeld., Projective Geometries Over Finite Fields<\/i>, ${ref.volume}<\/b> (1998).","DOI":"10.1093\/oso\/9780198502951.001.0001"},{"key":"key-10.3934\/amc.2020105-27","doi-asserted-by":"publisher","unstructured":"T. Ho, M. M\u00e9dard, R. Koetter, D. R. Karger, M. Effros, J. Shi, B. Leong.A random linear network coding approach to multicast, IEEE Trans. Inform. Theory<\/i>, 52<\/b> (2006), 4413-4430.","DOI":"10.1109\/TIT.2006.881746"},{"key":"key-10.3934\/amc.2020105-28","doi-asserted-by":"publisher","unstructured":"T. Ho, R. Koetter, M. M\u00e9dard, D. R. Karger and M. Effros, The benefits of coding over routing in a randomized setting, in Proceedings of the 2003 IEEE international symposium on information theory (ISIT 2003)<\/i>, Yokohama, Japan. IEEE, (2003), p442.","DOI":"10.1109\/ISIT.2003.1228459"},{"key":"key-10.3934\/amc.2020105-29","doi-asserted-by":"crossref","unstructured":"T. Honold, M. Kiermaier and S. Kurz, Partial spreads and vector space partitions, in Network Coding and Subspace Designs<\/i>, 131\u2013170, Signals Commun. Technol., Springer, Cham, 2018.","DOI":"10.1007\/978-3-319-70293-3_7"},{"key":"key-10.3934\/amc.2020105-30","doi-asserted-by":"publisher","unstructured":"A.-L. Horlemann-Trautmann.Message encoding and retrieval for spread and cyclic orbit codes, Des. Codes Cryptogr.<\/i>, 86<\/b> (2018), 365-386.","DOI":"10.1007\/s10623-017-0377-x"},{"key":"key-10.3934\/amc.2020105-31","doi-asserted-by":"crossref","unstructured":"A. L. Horlemann-Trautmann and J. Rosenthal, Constructions of constant dimension codes, in Network Coding and Subspace Designs<\/i>, 25\u201342, Signals Commun. Technol., Springer, Cham, 2018.","DOI":"10.1007\/978-3-319-70293-3_2"},{"key":"key-10.3934\/amc.2020105-32","doi-asserted-by":"crossref","unstructured":"B. Huppert, Endliche Gruppen I<\/i>, Springer-Verlag, Berlin-New York, 1967.","DOI":"10.1007\/978-3-642-64981-3"},{"key":"key-10.3934\/amc.2020105-33","doi-asserted-by":"publisher","unstructured":"W. M. Kantor.Linear groups containing a Singer cycle, J. Algebra<\/i>, 62<\/b> (1980), 232-234.","DOI":"10.1016\/0021-8693(80)90214-8"},{"key":"key-10.3934\/amc.2020105-34","doi-asserted-by":"publisher","unstructured":"B. C. Kestenband.Finite projective geometries that are incidence structures of caps, Linear Algebra Appl.<\/i>, 48<\/b> (1982), 303-313.","DOI":"10.1016\/0024-3795(82)90116-1"},{"key":"key-10.3934\/amc.2020105-35","doi-asserted-by":"publisher","unstructured":"A. Kohnert, S. Kurz.Construction of large constant-dimension codes with a prescribed minimum distance, Lecture Notes in Computer Science<\/i>, 5393<\/b> (2008), 31-42.","DOI":"10.1007\/978-3-540-89994-5_4"},{"key":"key-10.3934\/amc.2020105-36","doi-asserted-by":"publisher","unstructured":"R. K\u00f6tter, F. R. Kschischang.Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory<\/i>, 54<\/b> (2008), 3579-3591.","DOI":"10.1109\/TIT.2008.926449"},{"key":"key-10.3934\/amc.2020105-37","doi-asserted-by":"crossref","unstructured":"R. Lidl, H. Niederreiter., Finite Fields<\/i>, ${ref.volume}<\/b> (1997).","DOI":"10.1017\/CBO9780511525926"},{"key":"key-10.3934\/amc.2020105-38","doi-asserted-by":"publisher","unstructured":"K. Otal, F. \u00d6zbudak.Cyclic subspace codes via subspace polynomials, Des. Codes Cryptogr.<\/i>, 85<\/b> (2017), 191-204.","DOI":"10.1007\/s10623-016-0297-1"},{"key":"key-10.3934\/amc.2020105-39","doi-asserted-by":"crossref","unstructured":"K. Otal and F. \u00d6zbudak, Constructions of cyclic subspace codes and maximum rank distance codes, in Network Coding and Subspace Designs<\/i>, 43\u201366, Signals Commun. Technol., Springer, Cham, 2018.","DOI":"10.1007\/978-3-319-70293-3_3"},{"key":"key-10.3934\/amc.2020105-40","doi-asserted-by":"publisher","unstructured":"M. H. Poroch and A. A. Talebi, Product of symplectic groups and its cyclic orbit code, Discrete Math. Algorithms Appl.<\/i>, 11<\/b> (2019), 1950061, 25 pp.","DOI":"10.1142\/s1793830919500617"},{"key":"key-10.3934\/amc.2020105-41","doi-asserted-by":"publisher","unstructured":"N. Silberstein, A.-L. Trautmann.Subspace codes based on graph matchings, Ferrers diagrams, and pending blocks, IEEE Trans. Inform. Theory<\/i>, 61<\/b> (2015), 3937-3953.","DOI":"10.1109\/TIT.2015.2435743"},{"key":"key-10.3934\/amc.2020105-42","doi-asserted-by":"publisher","unstructured":"D. Silva, F. R. Kschischang, R. Koetter.A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory<\/i>, 54<\/b> (2008), 3951-3967.","DOI":"10.1109\/TIT.2008.928291"},{"key":"key-10.3934\/amc.2020105-43","doi-asserted-by":"publisher","unstructured":"J. Singer.A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc.<\/i>, 43<\/b> (1938), 377-385.","DOI":"10.1090\/S0002-9947-1938-1501951-4"},{"key":"key-10.3934\/amc.2020105-44","doi-asserted-by":"publisher","unstructured":"A.-L. Trautmann.Isometry and automorphisms of constant dimension codes, Adv. Math. Commun.<\/i>, 7<\/b> (2013), 147-160.","DOI":"10.3934\/amc.2013.7.147"},{"key":"key-10.3934\/amc.2020105-45","doi-asserted-by":"publisher","unstructured":"A.-L. Trautmann, F. Manganiello, M. Braun, J. Rosenthal.Cyclic orbit codes, IEEE Trans. Inf. Theory<\/i>, 59<\/b> (2013), 7386-7404.","DOI":"10.1109\/TIT.2013.2274266"},{"key":"key-10.3934\/amc.2020105-46","doi-asserted-by":"publisher","unstructured":"A.-L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - a new concept in the area of network coding, in Proc. IEEE Inf. Theory Workshop<\/i>, Dublin, Ireland, 2010, 1\u20134.","DOI":"10.1109\/CIG.2010.5592788"},{"key":"key-10.3934\/amc.2020105-47","unstructured":"D. E. Taylor, The Geometry of the Classical Groups<\/i>, Sigma Series in Pure Mathematics, 9. Heldermann Verlag, Berlin, 1992."},{"key":"key-10.3934\/amc.2020105-48","doi-asserted-by":"publisher","unstructured":"Z.-X. Wan, Geometry of matrices<\/i>, World Scientific Publishing Co. NJ, 1996.","DOI":"10.1142\/9789812830234"},{"key":"key-10.3934\/amc.2020105-49","doi-asserted-by":"publisher","unstructured":"B. Wu, Z. Liu.Linearized polynomials over finite fields revisited, Finite Fields Appl.<\/i>, 22<\/b> (2013), 79-100.","DOI":"10.1016\/j.ffa.2013.03.003"},{"key":"key-10.3934\/amc.2020105-50","doi-asserted-by":"publisher","unstructured":"S.-T. Xia, F.-W. Fu.Johnson type bounds on constant dimension codes, Des. Codes Cryptogr.<\/i>, 50<\/b> (2009), 163-172.","DOI":"10.1007\/s10623-008-9221-7"}],"container-title":["Advances in Mathematics of Communications"],"original-title":[],"link":[{"URL":"https:\/\/www.aimsciences.org\/article\/exportPdf?id=70f4a3e2-44a6-4abb-9be7-1a6ea05a15f8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,12]],"date-time":"2024-08-12T11:03:38Z","timestamp":1723460618000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/amc.2020105"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022]]},"references-count":50,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022]]}},"alternative-id":["1930-5346_2022_1_135"],"URL":"https:\/\/doi.org\/10.3934\/amc.2020105","relation":{},"ISSN":["1930-5346","1930-5338"],"issn-type":[{"type":"print","value":"1930-5346"},{"type":"electronic","value":"1930-5338"}],"subject":[],"published":{"date-parts":[[2022]]}}}