Permutations of zero-sumsets in a finite vector space

  • 1 Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123, Palermo, Italy
  • 2 Dipartimento di Ingegneria, Università degli Studi di Palermo, Viale delle Scienze, 90128, Palermo, Italy
Giovanni FalconeORCID iD: https://orcid.org/0000-0002-5210-5416 and Marco PavoneORCID iD: https://orcid.org/0000-0002-8674-5841

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.

  • [1]

    T. Beth, D. Jungnickel and H. Lenz, Design Theory, 2nd ed., Cambridge University, Cambridge, 1999.

  • [2]

    A. Caggegi, G. Falcone and M. Pavone, On the additivity of block designs, J. Algebraic Combin. 45 (2017), no. 1, 271–294.

    • Crossref
    • Export Citation
  • [3]

    A. Caggegi, G. Falcone and M. Pavone, Additivity of affine designs, J. Algebraic Combin. (2020), 10.1007/s10801-020-00941-8.

  • [4]

    C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, 2nd ed., CRC Press, Boca Raton, 2007.

  • [5]

    M. Kosters, The subset sum problem for finite abelian groups, J. Combin. Theory Ser. A 120 (2013), no. 3, 527–530.

    • Crossref
    • Export Citation
  • [6]

    J. Li and D. Wan, On the subset sum problem over finite fields, Finite Fields Appl. 14 (2008), no. 4, 911–929.

    • Crossref
    • Export Citation
  • [7]

    J. Li and D. Wan, Counting subset sums of finite abelian groups, J. Combin. Theory Ser. A 119 (2012), no. 1, 170–182.

    • Crossref
    • Export Citation
  • [8]

    M. B. Nathanson, Additive Number Theory, Grad. Texts in Math. 165, Springer, New York, 1996.

  • [9]

    M. Pavone, Subset sums and block designs in a finite vector space, manuscript.

  • [10]

    T. Tao and V. Vu, Additive Combinatorics, Cambridge Stud. Adv. Math. 105, Cambridge University, Cambridge, 2006.

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