Name
Title | ||
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Note: An empty interval in the spectrum of small weight codewords in the code from points and k-spaces of PG(n,q) |
Abstract | ||
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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^+^1-1q-1,2q^k[ in C"k(n,q)@?C"n"-"k(n,q)^@? which implies that there are no codewords with this weight in C"k(n,q)@?C"k(n,q)^@? if k=n/2. In particular, for the code C"n"-"1(n,q) of points and hyperplanes of PG(n,q), we exclude all codewords in C"n"-"1(n,q) with weight in the open interval ]q^n-1q-1,2q^n^-^1[. This latter result implies a sharp bound on the weight of small weight codewords of C"n"-"1(n,q), a result which was previously only known for general dimension for q prime and q=p^2, with p prime, p11, 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]. |
Year | DOI | Venue |
|---|---|---|
2009 | 10.1016/j.jcta.2008.09.004 | Journal of Combinatorial Theory Series A |
Keywords | Field | DocType |
weight distribution,open interval,incidence matrix,code c,l. storme,empty interval,m. lavrauw,codes cryptogr,l. fancsali,g. van de voorde,small weight codewords,projective space,linear code,spectrum | Prime (order theory),Discrete mathematics,Finite field,Combinatorics,Linear code,Hyperplane,Projective plane,Incidence matrix,Mathematics | Journal |
Volume | Issue | ISSN |
116 | 4 | 0097-3165 |
Citations | PageRank | References |
4 | 0.45 | 6 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| 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 |