Designs, Codes and Cryptography, 37 (2005), 211-214.
(t,m,s)-nets were defined by Niederreiter, based on earlier work by Sobol, in the context of quasi-Monte Carlo methods of numerical integration. Formulated in combinatorial/coding theoretic terms a binary linear (m-k,m,s)_{2}-net is a family of ks vectors in GF(2)^{m} satisfying certain linear independence conditions (s is the length, m the dimension and k the strength: certain subsets of k vectors must be linearly independent). Helleseth-Klove-Levenshtein recently constructed (2r-3,2r+2,2^{r}-1)_{2}-nets for every r. In this paper we give a direct and elementary construction for (2r-3,2r+2,2^{r}+1)_{2}-nets based on a family of binary linear codes of minimum distance 6.
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