Abstract
In Wu (Linear Algebra Appl 439:422–434 2013), the author constructed a binary code using the incidence matrix of conics consisting only of internal points with respect to a fixed conic versus internal points and studied geometric problems associated with this code. Inspired by that work, in this article, we construct conics consisting only of external points with respect to a conic for \(q\) odd. We study the intersection pattern of each of these conics with secant lines of the fixed conic, compute the dimension of the \({\mathbb F}_2\)-row space of the incidence matrix of the aforementioned conics and external points which provides us with the dimension of the associated binary code, and find the automorphism group of the binary code.
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References
Abatangelo V., Fisher J.C., Korchm\(\acute{a}\)ros G., Larato B.: On the mutual position of two irreducible conics in \(\text{ PG }(2, q)\), \(q\) odd. Adv. Geom. 11(4), 604–613 (2011).
Adams M., Wu J.: \(2\)-ranks of incidence matrices associated with conics in finite projective panes. Des. Codes Cryptogr. doi:10.1007/s10623-012-9772-5 (2014).
Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997).
Droms S., Mellinger K.E., Meyer C.: LDPC codes generated by conics in the classical projective plane. Des. Codes Cryptogr. 40, 343–356 (2006).
Gallager R.G.: Low-density parity-check codes. IRE Trans. Inform. Theory IT-8, 21–28 (1962).
Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Oxford University Press, Oxford (1998).
Hughes D.R., Piper F.C.: Projective Planes. Graduate Texts in Mathematics, vol. 6, Springer, New York (1973).
Kuo Y., Lin S., Fossorier M.P.C.: Low-density parity-check codes based on finite geometries: a rediscovery and new results. IEEE Trans. Inform. Theory 47, 2711–2736 (2001).
Madison A.L., Wu J.: On binary codes from conics in \(\text{ PG }(2, q)\). Eur. J. Combin. 33, 33–48 (2012).
Wu J.: Conics arising from internal points and their binary codes. Linear Algebra Appl. 439, 422–434 (2013).
Acknowledgments
Junhua Wu Research supported in part by NSF HBCU-UP Grant Award \(0929257\) at Lane College.
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Communicated by D. Ghinelli.
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Madison, A.L., Wu, J. Conics arising from external points and their binary codes. Des. Codes Cryptogr. 78, 473–491 (2016). https://doi.org/10.1007/s10623-014-0013-y
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DOI: https://doi.org/10.1007/s10623-014-0013-y