@article{61759,
  abstract     = {{Intersection distribution and non-hitting index are concepts introduced recently by Li and Pott as a new way to view the behaviour of a collection of finite field polynomials. With both an algebraic interpretation via the intersection of a polynomial with a set of lines, and a geometric interpretation via a (q+1)-set possessing an internal nucleus, the concepts have proved their usefulness as a new way to view various long-standing problems, and have applications in areas such as Kakeya sets. In this paper, by exploiting connections with diverse areas including the theory of algebraic curves, cyclotomy and the enumeration of irreducible polynomials, we establish new results and resolve various Open Problems of Li and Pott. We prove geometric results which shed new light on the relationship between intersection distribution and projective equivalence of polynomials, and algebraic results which describe and characterise the degree of Sf - the index of the largest non-zero entry in the intersection distribution of f. We provide new insights into the non-hitting spectrum, and show the limitations of the non-hitting index as a tool for characterisation. Finally, the benefits provided by the connections to other areas are evidenced in two short new proofs of the cubic case. }},
  author       = {{Klawuhn, Lukas-André Dominik and Huczynska, Sophie and Paterson, Maura}},
  journal      = {{Finite Fields and Their Applications}},
  publisher    = {{Elsevier}},
  title        = {{{The Intersection Distribution: New Results and Perspectives}}},
  doi          = {{10.1016/j.ffa.2026.102828}},
  volume       = {{114}},
  year         = {{2026}},
}

@article{60491,
  abstract     = {{We investigate generalisations of 1-factorisations and hyperfactorisations of the complete graph $K_{2n}$. We show that they are special subsets of the association scheme obtained from the Gelfand pair $(S_{2n},S_2 \wr S_n)$. This unifies and extends results by Cameron (1976) and gives rise to new existence and non-existence results. Our methods involve working in the group algebra $\mathbb{C}[S_{2n}]$ and using the representation theory of $S_{2n}$.}},
  author       = {{Klawuhn, Lukas-André Dominik and Bamberg, John}},
  journal      = {{Algebraic Combinatorics}},
  number       = {{3}},
  pages        = {{789--809}},
  title        = {{{On the association scheme of perfect matchings and their designs}}},
  doi          = {{10.5802/alco.490}},
  volume       = {{9}},
  year         = {{2026}},
}

@unpublished{66731,
  abstract     = {{We generalize intersection numbers for combinatorial designs to designs in finite meet-semilattices satisfying suitable regularity conditions. While designs in regular semilattices go back to Delsarte, our regularity assumptions are weaker than his and need not give rise to an association scheme. In this framework, we extend Mendelsohn's equations, prove a generalized Singleton bound with Steiner systems as equality cases, and determine the block intersection distribution at any block of a Steiner system. In particular, this distribution is independent of the chosen block.
  Specializing to several classical semilattice families, our results recover a number of well-known distributions in coding and design theory. In the Hamming and the $q$-Hamming (or bilinear forms) schemes, they give the local distance distributions of MDS and MRD codes, respectively. In the Johnson and $q$-Johnson (or Graßmann) schemes, they reproduce the block intersection distribution of classical and $q$-analog Steiner systems, equivalently the distance distribution of diameter-perfect constant-weight codes and diameter-perfect constant-dimension subspace codes. For the $q$-Johnson schemes, to the best of our knowledge, this result is new. As a further illustration, we apply our theory to designs of perfect matchings.
  Our approach provides a unified treatment of these cases in the strongest form known in the literature, determining the distribution relative to each individual block or codeword, without averaging and without linearity or additivity assumptions. Moreover, it identifies the natural double-counting objects underlying these distributions, leading to formulas in the regularity parameters of the semilattice and avoiding the more cumbersome expressions that arise in eigenvalue-based approaches via the ambient association scheme.}},
  author       = {{Kiermaier, Michael and Klawuhn, Lukas-André Dominik}},
  title        = {{{Intersection numbers for designs in regular semilattices}}},
  year         = {{2026}},
}

@unpublished{63384,
  abstract     = {{Two fundamental ways to represent a group are as permutations and as matrices. In this paper, we study linear representations of groups that intertwine with a permutation representation. Recently, D'Alconzo and Di Scala investigated how small the matrices in such a linear representation can be. The minimal dimension of such a representation is the \emph{linear dimension of the group action} and this has applications in cryptography and cryptosystems.

We develop the idea of linear dimension from an algebraic point of view by using the theory of permutation modules. We give structural results about representations of minimal dimension and investigate the implications of faithfulness, transitivity and primitivity on the linear dimension. Furthermore, we compute the linear dimension of several classes of finite primitive permutation groups. We also study wreath products, allowing us to determine the linear dimension of imprimitive group actions. Finally, we give the linear dimension of almost simple finite $2$-transitive groups, some of which may be used for further applications in cryptography. Our results also open up many new questions about linear representations of group actions.}},
  author       = {{Devillers, Alice and Giudici, Michael and Hawtin, Daniel R. and Klawuhn, Lukas-André Dominik and Morgan, Luke}},
  title        = {{{Linear dimension of group actions}}},
  year         = {{2025}},
}

@unpublished{56429,
  abstract     = {{It is known that the notion of a transitive subgroup of a permutation group
$P$ extends naturally to the subsets of $P$. We study transitive subsets of the
wreath product $G \wr S_n$, where $G$ is a finite abelian group. This includes
the hyperoctahedral group for $G=C_2$. We give structural characterisations of
transitive subsets using the character theory of $G \wr S_n$ and interpret such
subsets as designs in the conjugacy class association scheme of $G \wr S_n$. In
particular, we prove a generalisation of the Livingstone-Wagner theorem and
give explicit constructions of transitive sets. Moreover, we establish
connections to orthogonal polynomials, namely the Charlier polynomials, and use
them to study codes and designs in $C_r \wr S_n$. Many of our results extend
results about the symmetric group $S_n$.}},
  author       = {{Klawuhn, Lukas-André Dominik and Schmidt, Kai-Uwe}},
  booktitle    = {{arXiv:2409.20495}},
  pages        = {{38}},
  title        = {{{Transitivity in wreath products with symmetric groups}}},
  year         = {{2024}},
}