Title | ||
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An empty interval in the spectrum of small weight codewords in the code from points and k -spaces of PG(n,q)PG(n,q) |
Abstract | ||
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Let Ck(n,q) be the p-ary linear code defined by the incidence matrix of points and k-spaces in PG(n,q), q=ph, p prime, h⩾1. In this paper, we show that there are no codewords of weight in the open interval ]qk+1−1q−1,2qk[ in Ck(n,q)∖Cn−k(n,q)⊥ which implies that there are no codewords with this weight in Ck(n,q)∖Ck(n,q)⊥ if k⩾n/2. In particular, for the code Cn−1(n,q) of points and hyperplanes of PG(n,q), we exclude all codewords in Cn−1(n,q) with weight in the open interval ]qn−1q−1,2qn−1[. This latter result implies a sharp bound on the weight of small weight codewords of Cn−1(n,q), a result which was previously only known for general dimension for q prime and q=p2, with p prime, p>11, and in the case n=2, for q=p3, p⩾7 [K. Chouinard, On weight distributions of codes of planes of order 9, Ars Combin. 63 (2002) 3–13; V. Fack, Sz.L. Fancsali, L. Storme, G. Van de Voorde, J. Winne, Small weight codewords in the codes arising from Desarguesian projective planes, Des. Codes Cryptogr. 46 (2008) 25–43; M. Lavrauw, L. Storme, G. Van de Voorde, On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual, Des. Codes Cryptogr. 48 (2008) 231–245; M. Lavrauw, L. Storme, G. Van de Voorde, On the code generated by the incidence matrix of points and k-spaces in PG(n,q) and its dual, Finite Fields Appl. 14 (2008) 1020–1038]. |
Year | DOI | Venue |
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2009 | 10.1016/j.jcta.2008.09.004 | Journal of Combinatorial Theory, Series A |
Keywords | DocType | Volume |
Projective spaces,Linear Codes,Blocking sets,Small weight codewords | Journal | 116 |
Issue | ISSN | Citations |
4 | 0097-3165 | 0 |
PageRank | References | Authors |
0.34 | 0 | 4 |
Name | Order | Citations | PageRank |
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Michel Lavrauw | 1 | 69 | 17.63 |
Leo Storme | 2 | 197 | 38.07 |
Peter Sziklai | 3 | 41 | 6.94 |
van de voorde | 4 | 35 | 7.85 |