1. An empty interval in the spectrum of small weight codewords in the code from points and k-spaces of PG(n, q)
- Author
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Lavrauw, Michel, Storme, Leo, Sziklai, Peter, Van de Voorde, Geertrui, Mathematics, and Computational and Applied Mathematics Programme
- Subjects
Projective spaces ,Blocking sets ,k-spaces ,Linear Codes ,FOS: Mathematics ,Mathematics - Combinatorics ,linear code ,Combinatorics (math.CO) ,Small weight codewords - Abstract
Let C-k(n, q) be the p-ary linear code defined by the incidence matrix of points and k-spaces in PG(n, q), q = p(h), p prime, h >= 1. in this paper, we show that there are no codewords of weight in the open interval ]q(k+1)-1/q - 1, 2q(k)[ in C-k(n, q) \ Cn-k(n, q)(perpendicular to) which implies that there are no codewords with this weight in C-k(n, q) \ C-k(n, q)(perpendicular to) 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 ]q(n)-1/q - 1, 2q(n-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 = p(2), with p prime, p > 11, and in the case n = 2, for q = p(3), 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]. (C) 2009 Elsevier Inc. All rights reserved.
- Published
- 2012