arXiv subjects
Publications and source records attributed to Robert S. Coulter.
We classify the parabolic unitals in regular nearfield planes of odd order $q^2$ whose linear collineation group has the maximal size of $q^3-q$. We also establish a number of more general results concerning parabolic unitals in regular nearfield planes under weaker assumptions.
Hughes introduced the projective planes that bear his name in 1957 and they have since been studied extensively. However, until now, no polynomial representation of a planar ternary ring that represents them has been determined. In this paper, we rectify this omission by determining a reduced PTR polynomial for any Hughes plane defined over a regular nearfield. The polynomials obtained provide a new surprising connection: both the Catalan numbers and generalized Catalan numbers occur among the coefficients, depending on the representation. Since every PTR polynomial has connections with several classes of permutation polynomials, we obtain three new infinite classes of permutation polynomials as a consequence of our main result, and these, too, involve the Catalan numbers. The differential uniformity of new permutation polynomials is also determined.
We show that the graph of a bent function is a Salem set in an appropriate sense. We also establish a simple result that quantifies redundancies in the difference operators of a function, which applies to bent functions over fields of odd characteristic via their equivalence to perfect non-linear functions in that setting. We end by demonstrating, by entirely elementary means, that the distance between two distinct planar functions must be at least two.
Permutation resemblance measures the distance of a function from being a permutation. Here we show how to determine the permutation resemblance through linear integer programming techniques. We also present an algorithm for constructing feasible solutions to this integer program, and use it to prove an upper bound for permutation resemblance for some special functions. Additionally, we present a generalization of the linear integer program that takes a function on a finite group and determines a permutation with the lowest differential uniformity among those most resembling it.
Motivated by the problem of constructing bijective maps with low differential uniformity, we introduce the notion of permutation resemblance of a function, which looks to measure the distance a given map is from being a permutation. We prove several results concerning permutation resemblance and show how it can be used to produce low differentially uniform bijections. We also study the permutation resemblance of planar functions, which over fields of odd characteristic are known not to be bijections and to have the optimal differential uniformity.
We study the differential uniformity of the Wan-Lidl polynomials over finite fields. A general upper bound, independent of the order of the field, is established. Additional bounds are established in settings where one of the parameters is restricted. In particular, we establish a class of permutation polynomials which have differential uniformity at most 5 over fields of order $3\bmod 4$, irrespective of the field size. Computational results are also given.
We present an example of a result in graph theory that is used to obtain a result in another branch of mathematics. More precisely, we show that the isomorphism of certain directed graphs implies that some trinomials over finite fields have the same number of roots.
Let $G$ be a 3-partite graph with $k$ vertices in each part and suppose that between any two parts, there is no cycle of length four. Fischer and Matouusek asked for the maximum number of triangles in such a graph. A simple construction involving arbitrary projective planes shows that there is such a graph with $(1 - o(1)) k^{3/2} $ triangles, and a double counting argument shows that one cannot have more than $(1+o(1)) k^{7/4} $ triangles. Using affine planes defined by specific planar polynomials over finite fields, we improve the lower bound to $(1 - o(1)) k^{5/3}$.
We consider the case where a particular incidence structure splits into two substructures. The incidence structure in question was used previously by the authors to construct semi-biplanes $sbp(k^2,k)$ or $sbp(k^2/2,k)$. A complete description of the two substructures is obtained. We also show that none of the three semi-biplanes, $sbp(18,6)$, can be described using this construction.
We report on a recent implementation of Giesbrecht's algorithm for factoring polynomials in a skew-polynomial ring. We also discuss the equivalence between factoring polynomials in a skew-polynomial ring and decomposing $p^s$-polynomials over a finite field, and how Giesbrecht's algorithm is outlined in some detail by Ore in the 1930's. We end with some observations on the security of the Hidden Field Equation (HFE) cryptosystem, where $p$-polynomials play a central role.
We revisit the coordinatisation method for projective planes. First, we discuss how the behaviour of the additive and multiplicative loops can be described in terms of its action on the "vertical" line, and how this means one can coordinatise certain planes in an optimal sense. We then move to consider projective planes of prime power order only. Specifically, we consider how coordinatising planes of prime power order using finite fields as the underlying labelling set leads to some general restrictions on the form of the resulting planar ternary ring (PTR) when viewed as a trivariate polynomial over the field. We also consider the Lenz-Barlotti type of the plane being coordinatised, deriving further restrictions on the form of the PTR polynomial.
In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In particular, we show that any sum set must exhibit higher-order regularity and that an abelian sum set is necessarily a reversible difference set. We next develop several general construction techniques under the hypothesis that the over-riding group contains a normal subgroup of order 2. Finally, by exploiting properties of dihedral groups and Frobenius groups, several infinite classes of sum sets and partial sum sets are introduced.