{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T17:05:52Z","timestamp":1682615152212},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T00:00:00Z","timestamp":1644278400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T00:00:00Z","timestamp":1644278400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Universit\u00e4t Bayreuth"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2022,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The classification of unitals with parameters 2-(28,\u00a04,\u00a01) according to the 2-rank of their incidence matrices was initiated by McGuire, Tonchev and Ward, who proved that the 2-rank of any unital on 28 points is greater than or equal to 19, and up to isomorphism, there is a unique unital with 2-rank equal to 19. Jaffe and Tonchev investigated the next two 2-ranks, 20 and 21, and showed that there are no unitals on 28 points with 2-rank equal to 20, and there are exactly 4 isomorphism classes of unitals of rank 21. The subject of this paper is the classification of unitals having 2-rank 22, 23 and 24.<\/jats:p>","DOI":"10.1007\/s00200-022-00541-y","type":"journal-article","created":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T10:03:37Z","timestamp":1644314617000},"page":"903-913","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the classification of unitals on 28 points of low rank"],"prefix":"10.1007","volume":"33","author":[{"given":"Vladimir","family":"Tonchev","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alfred","family":"Wassermann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,2,8]]},"reference":[{"key":"541_CR1","doi-asserted-by":"publisher","unstructured":"Al-Azemi, A., Betten, A., Betten, D.: Unital designs with blocking sets. Discrete Appl. Math. 163, 102\u2013112 (2014). https:\/\/doi.org\/10.1016\/j.dam.2013.02.023(Optimal Discrete Structures and Algorithms ODSA 2010)","DOI":"10.1016\/j.dam.2013.02.023"},{"key":"541_CR2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139507660","volume-title":"Design Theory","author":"T Beth","year":"1999","unstructured":"Beth, T., Jungnickel, D., Lenz, H.: Design Theory, vol. 1, 2nd edn. Cambridge University Press, Cambridge (1999)","edition":"2"},{"issue":"1\u20133","key":"541_CR3","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/S0012-365X(02)00600-3","volume":"267","author":"A Betten","year":"2003","unstructured":"Betten, A., Betten, D., Tonchev, V.D.: Unitals and codes. Discrete Math. 267(1\u20133), 23\u201333 (2003). https:\/\/doi.org\/10.1016\/S0012-365X(02)00600-3","journal-title":"Discrete Math."},{"key":"541_CR4","volume-title":"Error-Correcting Linear Codes: Classification by Isometry and Applications (Algorithms and Computation in Mathematics)","author":"A Betten","year":"2006","unstructured":"Betten, A., Braun, M., Fripertinger, H., Kerber, A., Kohnert, A., Wassermann, A.: Error-Correcting Linear Codes: Classification by Isometry and Applications (Algorithms and Computation in Mathematics), 1st edn. Springer, Berlin (2006)","edition":"1"},{"key":"541_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/TIT.2021.3114280","volume":"1","author":"I Bouyukliev","year":"2021","unstructured":"Bouyukliev, I., Bouyuklieva, S., Kurz, S.: Computer classification of linear codes. IEEE Trans. Inf. Theory 1, 1 (2021). https:\/\/doi.org\/10.1109\/TIT.2021.3114280","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"12","key":"541_CR6","doi-asserted-by":"publisher","first-page":"4247","DOI":"10.1109\/TIT.2005.859291","volume":"51","author":"M Braun","year":"2005","unstructured":"Braun, M., Kohnert, A., Wassermann, A.: Optimal linear codes from matrix groups. IEEE Trans. Inf. Theory 51(12), 4247\u20134251 (2005). https:\/\/doi.org\/10.1109\/TIT.2005.859291","journal-title":"IEEE Trans. Inf. Theory"},{"key":"541_CR7","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/BFb0091018","volume-title":"Geometries and Groups","author":"AE Brouwer","year":"1981","unstructured":"Brouwer, A.E.: Some unitals on 28 points and their embeddings in projective planes of order 9. In: Aigner, M., Jungnickel, D. (eds.) Geometries and Groups, pp. 183\u2013188. Springer, Berlin (1981)"},{"key":"541_CR8","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/978-1-4613-1423-3_4","volume-title":"Codes, Designs and Geometry","author":"LL Carpenter","year":"1996","unstructured":"Carpenter, L.L.: Oval designs in desarguesian projective planes. In: Tonchev, V. (ed.) Codes, Designs and Geometry, pp. 47\u201355. Springer US, Boston (1996). https:\/\/doi.org\/10.1007\/978-1-4613-1423-3_4"},{"issue":"4","key":"541_CR9","doi-asserted-by":"publisher","first-page":"363","DOI":"10.3934\/amc.2009.3.363","volume":"3","author":"T Feulner","year":"2009","unstructured":"Feulner, T.: The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes. Adv. Math. Commun. 3(4), 363\u2013383 (2009)","journal-title":"Adv. Math. Commun."},{"issue":"1","key":"541_CR10","doi-asserted-by":"publisher","first-page":"153","DOI":"10.32917\/hmj\/1206137446","volume":"3","author":"N Hamada","year":"1973","unstructured":"Hamada, N.: On the $$p$$-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error correcting codes. Hiroshima Math. J. 3(1), 153\u2013226 (1973)","journal-title":"Hiroshima Math. J."},{"issue":"1","key":"541_CR11","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1023\/A:1008204420606","volume":"14","author":"DB Jaffe","year":"1998","unstructured":"Jaffe, D.B., Tonchev, V.D.: Computing linear codes and unitals. Des. Codes Cryptogr. 14(1), 39\u201352 (1998). https:\/\/doi.org\/10.1023\/A:1008204420606","journal-title":"Des. Codes Cryptogr."},{"issue":"2","key":"541_CR12","doi-asserted-by":"publisher","first-page":"121","DOI":"10.2478\/s12175-009-0113-8","volume":"59","author":"P Kaski","year":"2009","unstructured":"Kaski, P., \u00d6sterg\u00e5rd, P.: Classification of resolvable balanced incomplete block designs\u2014the unitals on 28 points. Math. Slovaca 59(2), 121\u2013136 (2009). https:\/\/doi.org\/10.2478\/s12175-009-0113-8","journal-title":"Math. Slovaca"},{"key":"541_CR13","volume-title":"The Art of Computer Programming, vol. 4: Combinatorial Algorithms, Fascicle 5","author":"DE Knuth","year":"2020","unstructured":"Knuth, D.E.: The Art of Computer Programming, vol. 4: Combinatorial Algorithms, Fascicle 5. Addison-Wesley, Boston (2020)"},{"issue":"57","key":"541_CR14","first-page":"259","volume":"37","author":"V Kr\u010dadinac","year":"2002","unstructured":"Kr\u010dadinac, V.: Steiner $$2$$-designs $${S}(2,4,28)$$ with nontrivial automorphisms. Glas. Mat. 37(57), 259\u2013268 (2002)","journal-title":"Glas. Mat."},{"key":"541_CR15","unstructured":"Kurz, S.: LinCode\u2014Computer Classification of Linear Codes (2019)"},{"key":"541_CR16","doi-asserted-by":"publisher","first-page":"256","DOI":"10.1016\/0021-8693(66)90014-7","volume":"3","author":"H L\u00fcneburg","year":"1966","unstructured":"L\u00fcneburg, H.: Some remarks concerning the Reed groups of type $$(G_2)$$. J. Algebra 3, 256\u2013259 (1966)","journal-title":"J. Algebra"},{"key":"541_CR17","unstructured":"Mackenzie, K.: Codes of Designs. Ph.D. thesis, University of Birmingham (1989)"},{"issue":"1","key":"541_CR18","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1023\/A:1008246022572","volume":"13","author":"G McGuire","year":"1998","unstructured":"McGuire, G., Tonchev, V.D., Ward, H.N.: Characterizing the Hermitian and Ree unitals on 28 points. Des. Codes Cryptogr. 13(1), 57\u201361 (1998). https:\/\/doi.org\/10.1023\/A:1008246022572","journal-title":"Des. Codes Cryptogr."},{"key":"541_CR19","doi-asserted-by":"publisher","first-page":"114","DOI":"10.1016\/j.dam.2020.08.026","volume":"288","author":"D Mez\u0151fi","year":"2021","unstructured":"Mez\u0151fi, D., Nagy, G.P.: New Steiner 2-designs from old ones by paramodifications. Discrete.Appl. Math. 288, 114\u2013122 (2021). https:\/\/doi.org\/10.1016\/j.dam.2020.08.026","journal-title":"Discrete.Appl. Math."},{"issue":"5","key":"541_CR20","doi-asserted-by":"publisher","first-page":"651","DOI":"10.1109\/TIT.1970.1054513","volume":"16","author":"SW Ng","year":"1970","unstructured":"Ng, S.W.: On Rudolph\u2019s majority-logic decoding algorithm (corresp.). IEEE Trans. Inf. Theory 16(5), 651\u2013652 (1970). https:\/\/doi.org\/10.1109\/TIT.1970.1054513","journal-title":"IEEE Trans. Inf. Theory"},{"key":"541_CR21","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/BF01388477","volume":"6","author":"T Penttila","year":"1995","unstructured":"Penttila, T., Royle, G.F.: Sets of type $$(m, n)$$ in the affine and projective planes of order nine. Des. Codes Cryptogr. 6, 229\u2013245 (1995)","journal-title":"Des. Codes Cryptogr."},{"issue":"2","key":"541_CR22","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1109\/TIT.1967.1053994","volume":"13","author":"LD Rudolph","year":"1967","unstructured":"Rudolph, L.D.: A class of majority logic decodable codes (corresp.). IEEE Trans. Inf. Theory 13(2), 305\u2013307 (1967). https:\/\/doi.org\/10.1109\/TIT.1967.1053994","journal-title":"IEEE Trans. Inf. Theory"},{"key":"541_CR23","unstructured":"Soicher, L.H.: The DESIGN package for GAP, Version 1.7 (2019). https:\/\/www.gap-system.org\/Packages\/design.html"},{"key":"541_CR24","doi-asserted-by":"publisher","first-page":"677","DOI":"10.1016\/S1570-7954(07)05005-X","volume-title":"Handbook of Combinatorial Designs, Chap. VI.1","author":"VD Tonchev","year":"2007","unstructured":"Tonchev, V.D.: Codes. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, Chap. VI.1, 2nd edn., pp. 677\u2013702. Hoboken, CRC (2007). https:\/\/doi.org\/10.1016\/S1570-7954(07)05005-X","edition":"2"},{"key":"541_CR25","unstructured":"The GAP\u00a0Group: GAP\u2014Groups, Algorithms, and Programming, Version 4.11.1 (2021). https:\/\/www.gap-system.org"},{"issue":"1\u20132","key":"541_CR26","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1007\/s10623-016-0243-2","volume":"84","author":"VD Tonchev","year":"2017","unstructured":"Tonchev, V.D.: On resolvable Steiner 2-designs and maximal arcs in projective planes. Des. Codes Cryptogr. 84(1\u20132), 165\u2013172 (2017). https:\/\/doi.org\/10.1007\/s10623-016-0243-2","journal-title":"Des. Codes Cryptogr."},{"key":"541_CR27","doi-asserted-by":"publisher","unstructured":"Tonchev, V.D., Wassermann, A.: On the classification of unitals on 28 points of low rank (2022). https:\/\/doi.org\/10.15495\/do_ubt_2065(accompanying data set)","DOI":"10.15495\/do_ubt_2065"},{"key":"541_CR28","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1023\/A:1013355015853","volume":"6","author":"A Wassermann","year":"2002","unstructured":"Wassermann, A.: Attacking the market split problem with lattice point enumeration. J. Comb. Optim. 6, 5\u201316 (2002)","journal-title":"J. Comb. Optim."},{"key":"541_CR29","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1007\/978-3-030-79987-8_2","volume-title":"Combinatorial Algorithms","author":"A Wassermann","year":"2021","unstructured":"Wassermann, A.: Search for combinatorial objects using lattice algorithms\u2014revisited. In: Flocchini, P., Moura, L. (eds.) Combinatorial Algorithms, pp. 20\u201333. Springer International Publishing, Cham (2021)"}],"container-title":["Applicable Algebra in Engineering, Communication and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-022-00541-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00200-022-00541-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-022-00541-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T16:06:30Z","timestamp":1668701190000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00200-022-00541-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,8]]},"references-count":29,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2022,12]]}},"alternative-id":["541"],"URL":"https:\/\/doi.org\/10.1007\/s00200-022-00541-y","relation":{},"ISSN":["0938-1279","1432-0622"],"issn-type":[{"value":"0938-1279","type":"print"},{"value":"1432-0622","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,8]]},"assertion":[{"value":"22 November 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 January 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 February 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}