Abstract
We study codewords of small weight in the codes arising from Desarguesian projective planes. We first of all improve the results of K. Chouinard on codewords of small weight in the codes arising from PG(2, p), p prime. Chouinard characterized all the codewords up to weight 2p in these codes. Using a particular basis for this code, described by Moorhouse, we characterize all the codewords of weight up to 2p + (p−1)/2 if p ≥ 11. We then study the codes arising from \(PG(2, q=q_0^3)\) . In particular, for q 0 = p prime, p ≥ 7, we prove that the codes have no codewords with weight in the interval [q + 2, 2q − 1]. Finally, for the codes of PG(2, q), q = p h, p prime, h ≥ 4, we present a discrete spectrum for the weights of codewords with weights in the interval [q + 2, 2q − 1]. In particular, we exclude all weights in the interval [3q/2, 2q − 1].
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References
Assmus E.F. Jr., Key J.D. (1992) Designs and their codes. Cambridge University Press, Cambridge
Ball S., Blokhuis A. (1996) On the size of a double blocking set in PG(2,q). Finite Fields Appl. 2, 125–137
Chouinard K.L.: Weight distributions of codes from planes. Ph.D Thesis, University of Virginia (2000).
Chouinard K.L. (2002) On weight distributions of codes of planes of order 9. Ars. Combin. 63, 3–13
McGuire G., Ward H.N. (1998) A determination of the weight enumerator of the code of the projective plane of order 5. Note Mat. 18(1): 71–99
McKay B.D.: nauty User’s Guide (Version 2.2) Computer Science Department, Australian National University (2004).
Moorhouse G.E. (1991) Bruck nets, codes, and characters of loops. Des. Codes Cryptogr. 1(1): 7–29
Polverino O. (1999) Small minimal blocking sets and complete k-arcs in PG(2,p 3). Discrete Math. 208/209: 469–476
Polverino O. (2000) Small blocking sets in PG(2,p 3). Des. Codes Cryptogr. 20: 319–324
Polverino O., Storme L. (2002) Small minimal blocking sets in PG(2,p 3). European J. Combin. 23: 83–92
Sachar H.: Error-correcting codes associated with finite planes. Ph.D Thesis, Lehigh University (1973).
Sloane N.J.A.: On-line Encyclopedia of Integer Sequences. http://www.research.att.com/~njas/sequences.
Sziklai P.: On small blocking sets and their linearity. J. Combin. Theory, Ser. A (submitted).
Szőnyi T. (1997) Blocking sets in Desarguesian affine and projective planes. Finite Fields Appl. 3, 187–202
Weisstein E.W.: Motzkin Number. http://mathworld.wolfram.com/MotzkinNumber.html.
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Communicated by J.D. Key.
Geertrui Van de Voorde research is supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen)
Joost Winne was supported by the Fund for Scientific Research - Flanders (Belgium).
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Fack, V., Fancsali, S.L., Storme, L. et al. Small weight codewords in the codes arising from Desarguesian projective planes. Des. Codes Cryptogr. 46, 25–43 (2008). https://doi.org/10.1007/s10623-007-9126-x
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DOI: https://doi.org/10.1007/s10623-007-9126-x