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On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual

Michel Lavrauw (UGent) , Leo Storme (UGent) and Geertrui Van de Voorde (UGent)
(2008) DESIGNS CODES AND CRYPTOGRAPHY. 48(3). p.231-245
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Abstract
In this paper, we study the p-ary linear code C(PG(n,q)), q = p(h), p prime, h >= 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2,q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar's lower bound (Geom Dedicata 8: 407 - 415, 1979).
Keywords
linear codes, projective spaces, blocking sets, small weight codewords

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MLA
Lavrauw, Michel, Leo Storme, and Geertrui Van de Voorde. “On the Code Generated by the Incidence Matrix of Points and Hyperplanes in PG(n,q) and Its Dual.” DESIGNS CODES AND CRYPTOGRAPHY 48.3 (2008): 231–245. Print.
APA
Lavrauw, M., Storme, L., & Van de Voorde, G. (2008). On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual. DESIGNS CODES AND CRYPTOGRAPHY, 48(3), 231–245.
Chicago author-date
Lavrauw, Michel, Leo Storme, and Geertrui Van de Voorde. 2008. “On the Code Generated by the Incidence Matrix of Points and Hyperplanes in PG(n,q) and Its Dual.” Designs Codes and Cryptography 48 (3): 231–245.
Chicago author-date (all authors)
Lavrauw, Michel, Leo Storme, and Geertrui Van de Voorde. 2008. “On the Code Generated by the Incidence Matrix of Points and Hyperplanes in PG(n,q) and Its Dual.” Designs Codes and Cryptography 48 (3): 231–245.
Vancouver
1.
Lavrauw M, Storme L, Van de Voorde G. On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual. DESIGNS CODES AND CRYPTOGRAPHY. 2008;48(3):231–45.
IEEE
[1]
M. Lavrauw, L. Storme, and G. Van de Voorde, “On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual,” DESIGNS CODES AND CRYPTOGRAPHY, vol. 48, no. 3, pp. 231–245, 2008.
@article{434881,
  abstract     = {In this paper, we study the p-ary linear code C(PG(n,q)), q = p(h), p prime, h >= 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2,q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar's lower bound (Geom Dedicata 8: 407 - 415, 1979).},
  author       = {Lavrauw, Michel and Storme, Leo and Van de Voorde, Geertrui},
  issn         = {0925-1022},
  journal      = {DESIGNS CODES AND CRYPTOGRAPHY},
  keywords     = {linear codes,projective spaces,blocking sets,small weight codewords},
  language     = {eng},
  number       = {3},
  pages        = {231--245},
  title        = {On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual},
  url          = {http://dx.doi.org/10.1007/s10623-008-9203-9},
  volume       = {48},
  year         = {2008},
}

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