A Family of Binary (t, m,s)-Nets of Strength 5

Abstract

(t,m,s)-Nets were defined by Niederreiter [Monatshefte fur Mathematik, Vol. 104 (1987) pp. 273–337], based on earlier work by Sobol’ [Zh. Vychisl Mat. i mat. Fiz, Vol. 7 (1967) pp. 784–802], 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 F m2 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 et al. [5] recently constructed (2r−3,2r+2,2r−1)2-nets for every r. In this paper, we give a direct and elementary construction for (2r−3,2r+2,2r+1)2-nets based on a family of binary linear codes of minimum distance 6.