The paper develops a construction for finding fully symmetric integration formulas of arbitrary degree 2k +t in n-space such that the number of evaluation points is O ((2n)k/k t), n-~. Formulas of degrees 3, 5, 7, 9, are relatively simple and are presented in detail. The method has been tested by obtaining some special formulas of degrees 7, 9 and 1 t but these are not presented here.
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