Cyclotomic Schemes Research Articles

Difference sets in pseudocyclic association schemes

Wilson (1972) proved that for fixed positive integers d and r such that 2d divides r(r−1), there exists a subset Y of size r of the finite field of order q such that the r(r−1) nonzero differences occurring from Y are evenly distributed to the dth cyclotomic classes provided that q is sufficiently large relative to d and r. On the other hand, Kajiura, Matsumoto and Okuda (2019) introduced the concept of difference sets in commutative association schemes in their study on a quasi-Monte Carlo method to approximate integrations over the point sets. Then, Wilson's result implies the existence of difference sets in cyclotomic schemes. In this paper, we generalize Wilson's result into pseudocyclic commutative association schemes and show the existence of difference sets in pseudocyclic association schemes with sufficiently large numbers of points.

On the separability of cyclotomic schemes over finite fields

It is proved that with finitely many possible exceptions, each cyclotomic scheme over a finite field is determined up to isomorphism by the tensor of 2 2 -dimensional intersection numbers; for infinitely many schemes, this result cannot be improved. As a consequence, the Weisfeiler–Leman dimension of a Paley graph or tournament is at most 3 3 with possible exception of several small graphs.

Strongly regular graphs arising from non-weakly regular bent functions

In this paper, we study two special subsets of a finite field of odd characteristics associated with non-weakly regular bent functions. We show that those subsets associated to non-weakly regular even bent functions in the GMMF class (see Cesmelioglu et al. Finite Fields Appl. 24, 105–117 2013) are never partial difference sets (PDSs), and are PDSs if and only if they are trivial subsets. Moreover, we analyze the two known sporadic examples of non-weakly regular ternary bent functions given in Helleseth and Kholosha (IEEE Trans. Inf. Theory 52(5), 2018–2032 2006, Cryptogr. Commun. 3(4), 281–291 2011). We observe that corresponding subsets are non-trivial partial difference sets. We show that they are the union of some cyclotomic cosets and so correspond to 2-class fusion schemes of a cyclotomic scheme. We also present a further construction giving non-trivial PDSs from certain p-ary functions which are not bent functions.

Modular adjacency algebras, standard representations, and p-ranks of cyclotomic association schemes

In this paper, we consider cyclotomic association schemes $$S = \mathrm {Cyc}(p^a, d)$$S=Cyc(pa,d). We focus on the adjacency algebra of S over algebraically closed fields K of characteristic p. If $$p\equiv 1 \pmod {d}$$pź1(modd), $$p\equiv -1\pmod {d}$$pź-1(modd), or $$d\in \{2,3,4,5,6\}$$dź{2,3,4,5,6}, we identify the adjacency algebra of S over K as a quotient of a polynomial ring over an admissible ideal. In several cases, we determine the indecomposable direct sum decomposition of the standard module of S. As a consequence, we are able to compute the p-rank of several specific elements of the adjacency algebra of S over K.

Nonsymmetric primitive translation schemes of prime power order

It is well-known that translation schemes of prime order are exactly the cyclotomic schemes. In this paper, we show that there do not exist any two-, three-, or four-class nonsymmetric primitive translation schemes of prime square order. On the other hand, we find new nonsymmetric four- and five-class association schemes from cyclotomy as fission schemes of certain symmetric three-class association schemes. Moreover, we provide an affirmative answer to the following question raised by Song (J Algebr Comb 5:47---55, 1996): are there any other two-class primitive association schemes that admit symmetrizable fission schemes besides the cyclotomic scheme of index 2 for $$q \equiv 5 \hbox { (mod 8)}$$ q � 5 (mod 8) ? To be more specific, we show that a certain two-class primitive association scheme in the finite field $${\mathbb F}_{37^3}$$ F 37 3 constructed by Feng and Xiang (J Comb Theory Ser A 119:245---256, 2012) admits a four-class fission scheme. This fission scheme is realized as a fusion scheme of the cyclotomic scheme of index 28.

Three-class association schemes from cyclotomy

We give three constructions of three-class association schemes as fusion schemes of the cyclotomic scheme, two of which are primitive.

Four-class skew-symmetric association schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we investigate 4-class skew-symmetric association schemes. In recent work by the first author it was discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S.Y. Song in 1996. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that none of 2-class Johnson schemes admits a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.

On generalised Paley graphs and their automorphism groups

The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see \cite{Paley}). They arise as the relation graphs of symmetric cyclotomic association schemes. However, their automorphism groups may be much larger than the groups of the corresponding schemes. We determine the parameters for which the graphs are connected, or equivalently, the schemes are primitive. Also we prove that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and we determine precisely the parameters for this to occur. We prove that in the connected, non-Hamming case, the automorphism group of a generalised Paley graph is a primitive group of affine type, and we find sufficient conditions under which the group is equal to the one-dimensional affine group of the associated cyclotomic association scheme. The results have been applied in \cite{LLP} to distinguish between cyclotomic schemes and similar twisted versions of these schemes, in the context of homogeneous factorisations of complete graphs.

A new example of non-amorphous association schemes

E. R. van Dam gave an example of primitive non-amorphous association schemes in which every nontrivial relation is a strongly regular graph, as a fusion scheme of the cyclotomic scheme of class $45$ on $¥GF(2^{12})$. The aim of this paper is to present a new example of primitive non-amorphous association schemes in which every nontrivial relation is a strongly regular graph, as a fusion scheme of the cyclotomic scheme of class $75$ on $¥GF(2^{20})$. We also propose an infinite family of parameters of association schemes containing both of these two examples.

Normal cyclotomic schemes over a finite commutative ring

We study cyclotomic association schemes over a finite commutative ring R with identity. The main interest for us is to identify the normal cyclotomic schemes C, i.e. those for which Aut(C) ≤ AL 1(R). The problem is reduced to the case when the ring R is local in which a necessary condition of normality in terms of the subgroup of R × defining C, is given. This condition is proved to be sufficient for a classof local rings including the Galois rings of odd characteristic.

On cyclotomic schemes over finite near-fields

We introduce a concept of cyclotomic association scheme over a finite near-field $\mathbb{K}$ . It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL(V), where V is the linear space associated with $\mathbb{K}$ . A sufficient condition on a cyclotomic scheme $\mathcal{C}$ that guarantee the inclusion $\mathrm{Aut}(\mathcal{C})\le \mathrm{A} \Gamma \mathrm{L}(1,\mathbb{F}),$ where $\mathbb{F}$ is a finite field with $|\mathbb{K}|$ elements, is given.

A construction of amorphous association scheme from a pseudo-cyclic association scheme

In this paper, we give a construction of an amorphous association scheme from a pseudo-cyclic symmetric association scheme. This construction is a generalization of the construction of strongly regular graphs due to Mathon. As an example we give an isomorphism of two amorphous association scheme derived from two cyclotomic schemes.

Separability Number and Schurity Number of Coherent Configurations

To each coherent configuration (scheme) ${\cal C}$ and positive integer $m$ we associate a natural scheme $\widehat{\cal C}^{(m)}$ on the $m$-fold Cartesian product of the point set of ${\cal C}$ having the same automorphism group as ${\cal C}$. Using this construction we define and study two positive integers: the separability number $s({\cal C})$ and the Schurity number $t({\cal C})$ of ${\cal C}$. It turns out that $s({\cal C})\le m$ iff ${\cal C}$ is uniquely determined up to isomorphism by the intersection numbers of the scheme $\widehat{\cal C}^{(m)}$. Similarly, $t({\cal C})\le m$ iff the diagonal subscheme of $\widehat{\cal C}^{(m)}$ is an orbital one. In particular, if ${\cal C}$ is the scheme of a distance-regular graph $\Gamma$, then $s({\cal C})=1$ iff $\Gamma$ is uniquely determined by its parameters whereas $t({\cal C})=1$ iff $\Gamma$ is distance-transitive. We show that if ${\cal C}$ is a Johnson, Hamming or Grassmann scheme, then $s({\cal C})\le 2$ and $t({\cal C})=1$. Moreover, we find the exact values of $s({\cal C})$ and $t({\cal C})$ for the scheme ${\cal C}$ associated with any distance-regular graph having the same parameters as some Johnson or Hamming graph. In particular, $s({\cal C})=t({\cal C})=2$ if ${\cal C}$ is the scheme of a Doob graph. In addition, we prove that $s({\cal C})\le 2$ and $t({\cal C})\le 2$ for any imprimitive 3/2-homogeneous scheme. Finally, we show that $s({\cal C})\le 4$, whenever ${\cal C}$ is a cyclotomic scheme on a prime number of points.

Primitive Commutative Association Schemes with a Non-symmetric Relation of Valency 3

Let (X, {Ri}0⩽i⩽d) be a primitive commutative association scheme. If there is a non-symmetric relation Ri with valency 3, then the cardinality of X is equal to either p or p2 where p is an odd prime. Moreover, if |X|=p then (X, {Ri}0⩽i⩽d) is isomorphic to a cyclotomic scheme.

Terwilliger Algebras of Cyclotomic Schemes and Jacobi Sums

We show that theT-module structure of a cyclotomic scheme is described in term of Jacobi sums. It holds that an irreducibleT-module of a cyclotomic scheme fails to have maximal dimension if and only if Jacobi sums satisfy certain kind of equations, which are of some number theoretical interest in themselves.

Classification of association schemes with 18 and 19 vertices

The isomorphism classes of association schemes with 18 and 19 vertices are classified. We obtain 95 isomorphism classes of association schemes with 18 vertices and denote the representatives of the isomorphism classes as subschemes of 7 association schemes. We obtain 7 isomorphism classes of association schemes with 19 vertices and six of them are cyclotomic schemes.

Automorphism Groups of BWD-codes

In this paper we characterize the automorphism group of a particular class of irreducible cyclic codes by means of an important theorem of Carlitz and McConnel. The BWD-codes (balanced weight distribution) have the remarkable property that each nonzero weight appears with the same multiplicity. We also show that an irreducible cyclic BWD-code withNnonzero weights is a subscheme withNclasses of the Hamming scheme isomorphic to a cyclotomic scheme.

The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in [3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let ? be a Galois ring of characteristic 4,X a cyclotomic scheme over ? with classD and ? the Terwilliger algebra ofX, then we show that most of the irreducible ?-modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible ?-modules using Jacobi sums.

Commutative Association Schemes Whose Symmetrizations Have Two Classes

If a symmetric association scheme of class two is realized as the symmetrization of a commutative association scheme, then it either admits a unique symmetrizable fission scheme of class three or four, or admits three fission schemes, two of which are class three and one is of class four. We investigate the classification problem for symmetrizable (commutative) association schemes of two-class symmetric association schemes. In particular, we give a classification of association schemes whose symmetrizations are obtained from completely multipartite strongly regular graphs in the notion of wreath product of two schemes. Also the cyclotomic schemes associated to Paley graphs and their symmetrizable fission schemes are discussed in terms of their character tables.

Cyclotomic schemes over finite rings

This paper is the first of two papers by both authors on the subject indicated in the title. (Left) (right) cyclotomic schemes are association schemes the relations R i of which are {( x, y) | y − x ϵ C i}, where the C i are the (left) (right) orbits of a given group of units of a finite ring. Special emphasis is put on the case of (left) (right) “admissible” rings in which case the eigen- values of the scheme can be determined in a simple manner and the left and the right schemes are isomorphic to each other's Tamaschke-Delsarte dual.