Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 12, 2020

Permutations of zero-sumsets in a finite vector space

  • Giovanni Falcone ORCID logo EMAIL logo and Marco Pavone ORCID logo
From the journal Forum Mathematicum

Abstract

In this paper, we consider a finite-dimensional vector space 𝒫 over the Galois field GF(p), with p being an odd prime, and the family kx of all k-sets of elements of 𝒫 summing up to a given element x. The main result of the paper is the characterization, for x=0, of the permutations of 𝒫 inducing permutations of k0 as the invertible linear mappings of the vector space 𝒫 if p does not divide k, and as the invertible affinities of the affine space 𝒫 if p divides k. The same question is answered also in the case where the elements of the k-sets are required to be all nonzero, and, in fact, the two cases prove to be intrinsically inseparable.


Communicated by Manfred Droste


Funding statement: This research was supported by the University of Palermo (FFR).

References

[1] T. Beth, D. Jungnickel and H. Lenz, Design Theory, 2nd ed., Cambridge University, Cambridge, 1999. 10.1017/CBO9780511549533Search in Google Scholar

[2] A. Caggegi, G. Falcone and M. Pavone, On the additivity of block designs, J. Algebraic Combin. 45 (2017), no. 1, 271–294. 10.1007/s10801-016-0707-5Search in Google Scholar

[3] A. Caggegi, G. Falcone and M. Pavone, Additivity of affine designs, J. Algebraic Combin. (2020), 10.1007/s10801-020-00941-8. 10.1007/s10801-020-00941-8Search in Google Scholar

[4] C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, 2nd ed., CRC Press, Boca Raton, 2007. 10.1201/9781420010541Search in Google Scholar

[5] M. Kosters, The subset sum problem for finite abelian groups, J. Combin. Theory Ser. A 120 (2013), no. 3, 527–530. 10.1016/j.jcta.2012.10.006Search in Google Scholar

[6] J. Li and D. Wan, On the subset sum problem over finite fields, Finite Fields Appl. 14 (2008), no. 4, 911–929. 10.1016/j.ffa.2008.05.003Search in Google Scholar

[7] J. Li and D. Wan, Counting subset sums of finite abelian groups, J. Combin. Theory Ser. A 119 (2012), no. 1, 170–182. 10.1016/j.jcta.2011.07.003Search in Google Scholar

[8] M. B. Nathanson, Additive Number Theory, Grad. Texts in Math. 165, Springer, New York, 1996. 10.1007/978-1-4757-3845-2Search in Google Scholar

[9] M. Pavone, Subset sums and block designs in a finite vector space, manuscript. Search in Google Scholar

[10] T. Tao and V. Vu, Additive Combinatorics, Cambridge Stud. Adv. Math. 105, Cambridge University, Cambridge, 2006. 10.1017/CBO9780511755149Search in Google Scholar

Received: 2019-08-21
Revised: 2020-06-26
Published Online: 2020-12-12
Published in Print: 2021-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 26.4.2024 from https://www.degruyter.com/document/doi/10.1515/forum-2019-0228/html
Scroll to top button