Academic Journal
Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order p^2
| العنوان: | Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order p^2 |
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| المؤلفون: | De Boeck, Maarten, Van de Voorde, Geertrui |
| المصدر: | DESIGNS CODES AND CRYPTOGRAPHY ; ISSN: 0925-1022 ; ISSN: 1573-7586 |
| سنة النشر: | 2023 |
| المجموعة: | Ghent University Academic Bibliography |
| مصطلحات موضوعية: | Mathematics and Statistics, Applied Mathematics, Computer Science Applications, Projective plane, Antipodal plane, (Dual) code of projective plane, Minimum weight |
| الوصف: | The minimum weight of the code generated by the incidence matrix of points versus lines in a projective plane has been known for over 50 years. Surprisingly, finding the minimum weight of the dual code of projective planes of non-prime order is still an open problem, even in the Desarguesian case. In this paper, we focus on the case of projective planes of order p(2), where p is prime, and we link the existence of small weight code words in the dual code to the existence of embedded subplanes and antipodal planes. In the Desarguesian case, we can exclude such code words by showing a more general result that no antipodal plane of order at least 3 can be embedded in a Desarguesian projective plane. Furthermore, we use combinatorial arguments to rule out the existence of code words in the dual code of points and lines of an arbitrary projective plane of order p(2), p prime, of weight at most 2p(2) - 2p + 4 using more than two symbols. In particular, this leads to the result that the dual code of the Desarguesian projective plane PG(2, p(2)), p >= 5, has minimum weight at least 2p(2) - 2p + 5. |
| نوع الوثيقة: | article in journal/newspaper |
| وصف الملف: | application/pdf |
| اللغة: | English |
| Relation: | https://biblio.ugent.be/publication/01GQ6J7BMDZWDFQWRECB967382; https://biblio.ugent.be/publication/01GQ6J7BMDZWDFQWRECB967382/file/01GQ6JBGHGTPMGW051Y9AC6JKA |
| DOI: | 10.1007/s10623-022-01131-2 |
| الاتاحة: | https://biblio.ugent.be/publication/01GQ6J7BMDZWDFQWRECB967382 http://hdl.handle.net/1854/LU-01GQ6J7BMDZWDFQWRECB967382 https://doi.org/10.1007/s10623-022-01131-2 https://biblio.ugent.be/publication/01GQ6J7BMDZWDFQWRECB967382/file/01GQ6JBGHGTPMGW051Y9AC6JKA |
| Rights: | No license (in copyright) ; info:eu-repo/semantics/openAccess |
| رقم الانضمام: | edsbas.A1A5AC90 |
| قاعدة البيانات: | BASE |
| DOI: | 10.1007/s10623-022-01131-2 |
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