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Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order p^2

Maarten De Boeck (UGent) and Geertrui Van de Voorde
(2023) DESIGNS CODES AND CRYPTOGRAPHY. 91(3). p.895-920
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Abstract
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.
Keywords
Applied Mathematics, Computer Science Applications, Projective plane, Antipodal plane, (Dual) code of projective plane, Minimum weight

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MLA
De Boeck, Maarten, and Geertrui Van de Voorde. “Embedded Antipodal Planes and the Minimum Weight of the Dual Code of Points and Lines in Projective Planes of Order P^2.” DESIGNS CODES AND CRYPTOGRAPHY, vol. 91, no. 3, 2023, pp. 895–920, doi:10.1007/s10623-022-01131-2.
APA
De Boeck, M., & Van de Voorde, G. (2023). Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order p^2. DESIGNS CODES AND CRYPTOGRAPHY, 91(3), 895–920. https://doi.org/10.1007/s10623-022-01131-2
Chicago author-date
De Boeck, Maarten, and Geertrui Van de Voorde. 2023. “Embedded Antipodal Planes and the Minimum Weight of the Dual Code of Points and Lines in Projective Planes of Order P^2.” DESIGNS CODES AND CRYPTOGRAPHY 91 (3): 895–920. https://doi.org/10.1007/s10623-022-01131-2.
Chicago author-date (all authors)
De Boeck, Maarten, and Geertrui Van de Voorde. 2023. “Embedded Antipodal Planes and the Minimum Weight of the Dual Code of Points and Lines in Projective Planes of Order P^2.” DESIGNS CODES AND CRYPTOGRAPHY 91 (3): 895–920. doi:10.1007/s10623-022-01131-2.
Vancouver
1.
De Boeck M, Van de Voorde G. Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order p^2. DESIGNS CODES AND CRYPTOGRAPHY. 2023;91(3):895–920.
IEEE
[1]
M. De Boeck and G. Van de Voorde, “Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order p^2,” DESIGNS CODES AND CRYPTOGRAPHY, vol. 91, no. 3, pp. 895–920, 2023.
@article{01GQ6J7BMDZWDFQWRECB967382,
  abstract     = {{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.}},
  author       = {{De Boeck, Maarten and Van de Voorde, Geertrui}},
  issn         = {{0925-1022}},
  journal      = {{DESIGNS CODES AND CRYPTOGRAPHY}},
  keywords     = {{Applied Mathematics,Computer Science Applications,Projective plane,Antipodal plane,(Dual) code of projective plane,Minimum weight}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{895--920}},
  title        = {{Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order p^2}},
  url          = {{http://doi.org/10.1007/s10623-022-01131-2}},
  volume       = {{91}},
  year         = {{2023}},
}
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