Geodesic
Minimal cubature formulae of an even degree for the 2-torus
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 5 (2002) no. 3, pp. 267-274.

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In this paper, we derive the minimal even degree formulas for the 2-torus in the trigonometric case. All such formulas are obtained by solving several matrix equations. As far as we know, this is the first approach to determine all formulae of this type. Computational results by using a Computer Algebra System are presented. They verify that up to degree 30 there is only one minimal formula of even degree (and its dual) if one node is fixed. In all the cases computed, it turned out that the known lattice rules of rank 1 are the only minimal formulas. 
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M. V. Noskov; H. J. Schmid. Minimal cubature formulae of an even degree for the 2-torus. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 5 (2002) no. 3, pp. 267-274. http://geodesic.mathdoc.fr/item/SJVM_2002_5_3_a5/

[1] Osipov N. N., “On minimal cubatureules with given trigonometric exactness in 2 dimensions”, Cubature Formulae and their Applications, Ufa, 1996, 52–60 (in Russian)

[2] Noskov M. V., “Formulae for the approximate integration of periodic functions”, Metody Vychisl., 15 (1988), 19–22 (in Russian) | MR | Zbl

[3] Osipov N. N., “About families minimal cubature rules with even trigonometrical exactness in 2 dimensions”, Cubature Formulae and their Appl., Ufa, 1996, 61–63 (in Russian)

[4] Schmid H. J., “Interpolatorische Kubaturformeln”, Diss. Math., 220, 1983, 1–122 (in German) | MR