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On the Existence of Agievich-Primitive Partitions

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Abstract

We prove that for any positive integer \( m \) there exists a smallest positive integer \( N=N_q(m) \) such that for \( n>N \) there exist no Agievich-primitive partitions of the space \( {\bf F}_q^n \) into \( q^m \) affine subspaces of dimension \( n-m \). We give lower and upper bounds on the value \( N_q(m) \) and prove that \( N_q(2)=q+1 \). Results of the same type for partitions into coordinate subspaces are established.

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Notes

  1. The inner product is also very often called a scalar product, but over a finite field or ring, such a product is not scalar in the full sense, since the “square is equal to zero only on a zero vector” property does not hold.

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ACKNOWLEDGMENTS

The author is grateful to colleagues from Lomonosov Moscow State University and Sobolev Institute of Mathematics for meaningful discussions and to the anonymous referee for useful remarks that helped improving the quality of the paper.

Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation as part of the program for the Moscow Center for Fundamental and Applied Mathematics under agreement no. 075–15–2022–284.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    Yu. V. Tarannikov

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    Yu. V. Tarannikov

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  1. Yu. V. Tarannikov
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Correspondence to Yu. V. Tarannikov.

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Translated by V. Potapchouck

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Tarannikov, Y.V. On the Existence of Agievich-Primitive Partitions. J. Appl. Ind. Math. 16, 809–820 (2022). https://doi.org/10.1134/S1990478922040202

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