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A 2- (v , k , λ) design is additive (or strongly additive) if it is possible to embed it in a suitable abelian group G in such a way that its block set is contained in (or coincides with) the set of all zero-sum k-subsets of its point set. Explicit results on the additivity or strong additivity of symmetric designs and subspace 2-designs are presented. In particular, the strong additivity of PG d (n , q) , which was known to be additive only for q = 2 or d = n - 1 , is always established. [ABSTRACT FROM AUTHOR]

Based on the packing density of cross-polytopes in R n , more than 50 years ago Golomb and Welch proved that the packing density of Lee spheres in Z n must be strictly smaller than 1 provided that the radius r of the Lee sphere is large enough compared with n, which implies that there is no perfect Lee code for the corresponding parameters r and n. In this paper, we investigate the lattice packing density of Lee spheres with fixed radius r for infinitely many n. First we present a method to verify the nonexistence of the second densest lattice packing of Lee spheres of radius 2. Second, we consider the constructions of lattice packings with density δ n → 2 r (2 r + 1) r ! as n → ∞ . When r = 2 , the packing density can be improved to δ n → 2 3 as n → ∞ . [ABSTRACT FROM AUTHOR]

For a countable discrete amenable group G, it turns out that for any subsets H of G and E of Z with positive densities, there exists k ∈ N which depends only on the densities of H and E such that G k ⊂ (H · H - 1) E - E . [ABSTRACT FROM AUTHOR]

Device-to-Device (D2D) communication is proposed in cellular network as a vital technology to enhance spectral efficiency and system capacity, while reducing the energy consumption and the latency. Enabling D2D communications in multi-cell OFDMA systems poses two majors challenges: First, D2D equipements are characterizd by limited capacity of battery. Second, D2D devices inevitably suffer from co-channel interference (CCI) at cell edge. This paper addresses the mobility control (MC) for D2D in downlink multi-cell OFDMA system with cooperative relaying and Fractional Frequency Reuse (FFR) scheme. FFR and cooperative relaying schemes are proposed to improve energy efficiency (EE) and avoid CCI. In the edge region, according to a difference set and using the sectorization technique, the Frequency Reuse Factor (FRF) of 7/3 and 7/4 have been applied. The deployment of FFR with Amplify and Forward fixed relays improve significantly the system performance of FRF of 1 and 3. As shown in simulations results, the proposed cooperative MC scheme with FRF of 7/3 and 7/4 provides an improvement in term of EE nears 1 , 2. 10 6 bits/s/J and 1 , 3. 10 6 bits/s/J respectively in comparison with non-cooperative scheme with FRF=3. [ABSTRACT FROM AUTHOR]

Difference systems of sets (DSSs) are combinatorial structures introduced by Levenshtein, which are a generalization of cyclic difference sets and arise in connection with code synchronization. In this paper, we describe four direct constructions of optimal DSSs from finite projective geometries and present a recursive construction of DSSs by extending the known construction. As a consequence, new infinite families of optimal DSSs can be obtained. [ABSTRACT FROM AUTHOR]

The concept of a modular difference set was originally motivated by the cognate notion of modular Hadamard matrices, which have been researched extensively. We initiate the study of the repetition-parameter set in a modular difference set, and we relate the repetition-parameter set to integer partitions and Diophantine equations. By example, we show how a computational study of integer partitions can improve the upper bound on the size of such repetition-parameter set. All previously known examples of modular difference sets in a direct product of groups are concerned with a product of just two groups. We present new constructions of modular difference sets in a direct product of n groups. These new constructions suggest that the size of the repetition-parameter set is intimately related to the group's structure. A generalization of difference sets, partial difference sets, and relative difference sets, modular difference sets have been used to construct both modular symmetric designs and equiangular tight frames in finite fields. [ABSTRACT FROM AUTHOR]

In the paper, we develop further the properties of Schur rings over infinite groups, with particular emphasis on the virtually cyclic group Z × Z p , where p is a prime. We provide structure theorems for primitive sets in these Schur rings. In the case of Fermat and safe primes, a complete classification theorem is proven, which states that all Schur rings over Z × Z p are traditional. We also draw analogs between Schur rings over Z × Z p and partitions of difference sets over Z p . [ABSTRACT FROM AUTHOR]

We extend the concept of graph representations modulo integers introduced by Erdös and Evans to graph representations over finite rings and generalize it to representations of signed graphs. We introduce several representation numbers and product dimensions of graphs and signed graphs and compute these quantities for a few special classes of signed graphs. [ABSTRACT FROM AUTHOR]

We study finite groups G having a subgroup H and D ⊂ G \ H such that (i) the multiset { x y - 1 : x , y ∈ D } has every element that is not in H occur the same number of times (such a D is called a relative difference set); (ii) G = D ∪ D (- 1) ∪ H ; (iii) D ∩ D (- 1) = ∅ . We show that | H | = 2 , that H is central and that G is a group with a single involution. We also show that G cannot be abelian. We give infinitely many examples of such groups, including certain dicyclic groups, by using results of Schmidt and Ito. [ABSTRACT FROM AUTHOR]

Let G be a finite group and f : G → C be a function. For a non-empty finite subset Y ⊂ G , let I Y (f) denote the average of f over Y. Then, I G (f) is the average of f over G. Using the decomposition of f into irreducible components of C G as a representation of G × G , we define non-negative real numbers V(f) and D(Y), each depending only on f, Y, respectively, such that an inequality of the form | I G (f) - I Y (f) | ≤ V (f) · D (Y) holds. We give a lower bound of D(Y) depending only on # Y and # G . We show that the lower bound is achieved if and only if # { (x , y) ∈ Y 2 ∣ x - 1 y ∈ [ a ] } / # [ a ] is independent of the choice of the conjugacy class [ a ] ⊂ G for a ≠ 1 . We call such a Y ⊂ G as a pre-difference set in G, since the condition is satisfied if Y is a difference set. If G is abelian, the condition is equivalent to that Y is a difference set. We found a non-trivial pre-difference set in the dihedral group of order 16, where no non-trivial difference set exists. The pre-difference sets in non-abelian groups of order 16 are classified. A generalization to commutative association schemes is also given. [ABSTRACT FROM AUTHOR]

We make the following three observations regarding a question popularized by Katznelson: is every subset of Z which is a set of Bohr recurrence also a set of topological recurrence? (i) If G is a countable abelian group and E ⊆ G is an I 0 set, then every subset of E - E which is a set of Bohr recurrence is also a set of topological recurrence. In particular every subset of { 2 n - 2 m : n , m ∈ N } which is a set of Bohr recurrence is a set of topological recurrence. (ii) Let Z ω be the direct sum of countably many copies of Z with standard basis E. If every subset of (E - E) - (E - E) which is a set of Bohr recurrence is also a set of topological recurrence, then every subset of every countable abelian group which is a set of Bohr recurrence is also a set of topological recurrence. (iii) Fix a prime p and let F p ω be the direct sum of countably many copies of Z / p Z with basis (e i) i ∈ N . If for every p-uniform hypergraph with vertex set N and edge set F having infinite chromatic number, the Cayley graph on F p ω determined by { ∑ i ∈ F e i : F ∈ F } has infinite chromatic number, then every subset of F p ω which is a set of Bohr recurrence is a set of topological recurrence. [ABSTRACT FROM AUTHOR]

The difference set Δ s , t of two (nondeterministic, in general) transducers s , t is the set of all input words for which the output sets of the two transducers are not equal. When the two transducers realize homomorphisms, their difference set is the complement of the well known equality set of the two homomorphisms. However, we show that transducer difference sets result in Chomsky-like classes of languages that are different than the classes resulting from equality sets. We also consider the following word problem: given transducers s , t and input w , tell whether the output sets s (w) and t (w) are different. In general the problem is PSPACE -complete, but it becomes NP -complete when at least one of the given transducers has finite outputs. We also provide a PRAX (polynomial randomized approximation) algorithm for the word problem as well as for the NFA (in)equivalence problem. Our presentation of PRAX algorithms improves the original presentation. • The difference set of two transducers is the set of all input words for which the corresponding output sets are not equal. • We show that transducer difference sets result in language classes that are different than classes resulting from equality sets. • We also consider the following word problem: given transducers s , t and input w , tell whether s (w) and t (w) are different. • The problem is PSPACE-complete, but it becomes NP-complete when at least one of the given transducers has finite outputs. • We also provide polynomial randomized approximation (PRAX) algorithms for the word problem and for the NFA equivalence problem. • Our presentation of PRAX algorithms improves the original presentation. [ABSTRACT FROM AUTHOR]

Let $G$ be an additive group of order $v$ . A $k$ -element subset $D$ of $G$ is called a $(v, k, \lambda, t)$ -almost difference set if the expressions $g-h$ , for $g$ and $h$ in $D$ , represent $t$ of the non-identity elements in $G$ exactly $\lambda $ times and every other non-identity element $\lambda + 1$ times. Almost difference sets are highly sought after as they can be used to produce functions with optimal nonlinearity, cyclic codes, and sequences with three-level autocorrelation. A set of positive integers $A$ is called a Golomb ruler if the difference between two distinct elements of $A$ are different. In this paper, we use Singer type Golomb rulers to construct new families of almost difference sets. Additionally, we constructed 2-adesigns from these almost difference sets.

We demonstrate that every difference set in a finite Abelian group is equivalent to a certain 'regular' covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $. This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets. [ABSTRACT FROM AUTHOR]

We show that if $\mathcal {A}\subset \{1,\ldots,N\}$ has no solutions to $a-b=n^2$ with $a,b\in \mathcal {A}$ and $n\geq 1$ , then \[ \lvert \mathcal{A}\rvert \ll \frac{N}{(\log N)^{c\log\log \log N}} \] for some absolute constant $c>0$. This improves upon a result of Pintz, Steiger, and Szemerédi. [ABSTRACT FROM AUTHOR]

Boolean bent functions have been introduced by Rothaus in 1966, bent functions in odd characteristic were first considered in 1985 by Kumar, Scholtz, and Welch. Two books on bent functions and some surveys on bent functions and related topics mainly deal with the Boolean case. In this survey, we focus on bent and vectorial bent functions in odd characteristic. Lately, one can observe increasing interest in the bentness of functions from elementary abelian into cyclic groups. Following this development, we also survey recent results on this class of functions. [ABSTRACT FROM AUTHOR]

This paper aims to propose a new design known as array cyclic (k*, 4)-cycle design. The development of this design is by forming a combination between a cyclic (k1, k2)-cycle system and a near-two-factor that will be constructed. To do so, we need to introduce a new cycle system, namely a (3*, 4)-cycle system. Thereafter, we present an analysis for the case, v ≡ 8, 4 (mod 12). [ABSTRACT FROM AUTHOR]

Spread and partial spread constructions are the most powerful bent function constructions. A large variety of bent functions from a 2m-dimensional vector space V 2 m (p) over F p into F p can be generated, which are constant on the sets of a partition of V 2 m (p) obtained with the subspaces of the (partial) spread. Moreover, from spreads one obtains not only bent functions between elementary abelian groups, but bent functions from V 2 m (p) to B, where B can be any abelian group of order p k , k ≤ m . As recently shown (Meidl, Pirsic 2021), partitions from spreads are not the only partitions of V 2 m (2) , with these remarkable properties. In this article we present first such partitions—other than (partial) spreads—which we call bent partitions, for V 2 m (p) , p odd. We investigate general properties of bent partitions, like number and cardinality of the subsets of the partition. We show that with bent partitions we can construct bent functions from V 2 m (p) into a cyclic group Z p k . With these results, we obtain the first constructions of bent functions from V 2 m (p) into Z p k , p odd, which provably do not come from (partial) spreads. [ABSTRACT FROM AUTHOR]

This paper addresses the question of how intercalates occur in the two known infinite families of triple arrays, the Paley triple arrays constructed in 2005 by Preece et al., and the Triple arrays from difference sets in 2017 by Nilson and Cameron. The main reason for doing this is that the number of such embedded Latin squares is often used when checking whether two arrays are isotopic or not. We determine sharp bounds for the number and density of intercalates for the main subclasses of these families respectively. We also prove the existence of an infinite family of triple arrays in which every two occurrences of an entry lie in an intercalate. [ABSTRACT FROM AUTHOR]

The fact that the groups Z 2 m × Z 2 m contain difference sets was first established by induction by Jim Davis in his Virginia dissertation of 1987. Later that year we gave a direct construction for a very large family of highly structured inequivalent difference sets in these groups. In this paper, we give a proof of our result which we presented long ago in a colloquium at Wright State University, but which has never been published. While the proof lays bare the rich structure of the difference sets, its utter simplicity suggests that it might just be from the book. We also provide some historical background and elaborate on some of the structural properties of these difference sets which have motivated some recent progress toward the goal of classifying all hadamard 2-groups. [ABSTRACT FROM AUTHOR]

Parametric relations of General Difference sets have been obtained for the first time. The relationships between two General Difference Sets have been identified that correspond to the same circulant partial Hadamard matrix r − H (k × 2 k). It has been shown that the existence of circulant partial Hadamard matrices of type r − H (k × 2 k) reflects the existence of ϕ (2 k) circular partial Hadamard matrices of the same type. The value of λ k in (2 k , d ; λ 1 , λ 2 , ... , λ (2 k − 1) ) -General Difference Set corresponds to the circulant partial Hadamard matrix r − H (k × 2 k) obtained. [ABSTRACT FROM AUTHOR]

Let q be a prime of the form q=nN +1 for integers n≥1 and N>1. For q<105, we show that difference sets in the additive group of the field GF(q) are obtained from unions of cyclotomic classes of orders N=12, 20 and 24 and determine all such unions using a computer search. We then determine if the difference sets are equivalent to known cyclotomic or modified cyclotomic quadratic, quartic, sextic, or octic difference sets or their complements. This fills the gaps in the literature on the existence of difference sets from unions of cyclotomic classes for the specified orders. In addition, we extend Baumert and Fredricksen’s 1967 work on the construction of all inequivalent (127,63,31)-difference sets from unions of 18th-cyclotomic classes of GF(127) by constructing six inequivalent (127,64,32)-difference sets with zero added from unions of cyclotomic classes of order N=18. [ABSTRACT FROM AUTHOR]

Codebooks with low-coherence have wide utilization in code division multiple access (CDMA) communications,quantum information theory,compressed sensing and so on.In order to expand the number of codebooks,the restrictions on the transformation matrix were relaxed.Based on the Zadoff-Chu matrix,new codebooks were constructed using the difference set,almost difference set,and finite field character sum.The proposed codebooks were optimal or near optimal according to the Welch bound or Levenstein bound.Through experimental simulation,it is found that the deterministic measurement matrices constructed using these codebooks also have good performance in the process of compressed sensing.

We study finite groups G having a normal subgroup H and D ⊂ G \ H , D ∩ D - 1 = ∅ , such that the multiset { x y - 1 : x , y ∈ D } has every non-identity element occur the same number of times (such a D is called a DRAD difference set). We show that there are no such groups of order 4 p 2 , where p is an odd prime. [ABSTRACT FROM AUTHOR]

An equiangular tight frame (ETF) is a sequence of vectors in a Hilbert space that achieves equality in the Welch bound and so has minimal coherence. More generally, an equichordal tight fusion frame (ECTFF) is a sequence of equi-dimensional subspaces of a Hilbert space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every ECTFF is a type of optimal Grassmannian code, that is, an optimal packing of equi-dimensional subspaces of a Hilbert space. We construct ECTFFs by exploiting new relationships between known ETFs. Harmonic ETFs equate to difference sets for finite abelian groups. We say that a difference set for such a group is "paired" with a difference set for its Pontryagin dual when the corresponding subsequence of its harmonic ETF happens to be an ETF for its span. We show that every such pair yields an ECTFF. We moreover construct an infinite family of paired difference sets using quadratic forms over the field of two elements. Together this yields two infinite families of real ECTFFs. [ABSTRACT FROM AUTHOR]

In the previous work, we introduce a notion of pre-difference sets in a finite group G defined by weaker conditions than the difference sets. In this paper we gave a construction of a pre-difference set in G = N A with A an abelian subgroup and N a subgroup satisfying N ∩ A = { e } , from a difference set in N × A . This gives a (16, 6, 2) pre-difference set in D 16 and a (27, 13, 6) pre-difference set in UT(3, 3), where no non-trivial difference sets exist. We also give a product construction of pre-difference sets similar to Kesava Menon construction, which provides infinite series of pre-difference sets that are not difference sets. We show some necessary conditions for the existence of a pre-difference set in a group with index 2 subgroup. For the proofs, we use a rather simple framework "relation partitions," which is obtained by dropping an axiom from association schemes. Most results are proved in that frame work. [ABSTRACT FROM AUTHOR]

A (v,Γ,λ)difference graph in an additive group G of order v is a copy of an abstract simple graph Γ with vertices in G such that the list of all possible differences between adjacent vertices covers every non-zero element of G exactly λ times. It is proved here that a (v,Γ,λ)difference graph cannot exist if v≡2(mod4)and Γ is Eulerian of odd size. As a consequence, we are able to exhibit infinitely many counterexamples to a recent conjecture proposed M. Buratti, A. Nakic and A. Wassermann.

We provide some applications of a polynomial criterion for difference sets. These include counting the difference sets with specified parameters in terms of Hilbert functions, in particular a count of bent functions. We also consider the question about the bentness of certain Boolean functions introduced by Carlet when the C-condition introduced by him doesn't hold. [ABSTRACT FROM AUTHOR]

Using group rings and characters as in the theory of abelian difference sets, some nonexistence results for strong difference families are provided. Existences of strong difference families with base block size 3≤k≤9 are discussed. Via strong difference families, eight 2‐(v,k,λ) designs whose existences were unknown are constructed for (v,k,λ)∈{(3417,8,1),(3753,8,1),(2665,9,1),(3817,9,1),(4393,9,1),(7353,9,1),(12537,9,1),(345,9,3)}. [ABSTRACT FROM AUTHOR]

Let PG(Fqv) be the (v−1)‐dimensional projective space over Fq and let Γ be a simple graph of order qk−1q−1 for some k. A 2−(v,Γ,λ) design over Fq is a collection ℬ of graphs (blocks) isomorphic to Γ with the following properties: the vertex set of every block is a subspace of PG(Fqv); every two distinct points of PG(Fqv) are adjacent in exactly λ blocks. This new definition covers, in particular, the well‐known concept of a 2−(v,k,λ) design over Fq corresponding to the case that Γ is complete. In this study of a foundational nature we illustrate how difference methods allow us to get concrete nontrivial examples of Γ‐decompositions over F2 or F3 for which Γ is a cycle, a path, a prism, a generalized Petersen graph, or a Moebius ladder. In particular, we will discuss in detail the special and very hard case that Γ is complete and λ=1, that is, the Steiner 2‐designs over a finite field. Also, we briefly touch the new topic of near resolvable 2−(v,2,1) designs over Fq. This study has led us to some (probably new) collateral problems concerning difference sets. Supported by multiple examples, we conjecture the existence of infinite families of Γ‐decompositions over a finite field that can be obtained by suitably labeling the vertices of Γ with the elements of a Singer difference set. [ABSTRACT FROM AUTHOR]

A (v , k , λ) symmetric design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block. The designs associated to the symplectic difference sets introduced by Kantor (J Algebra 33:43–58, 1975) have the SDP. Parker (J Comb Theory Ser A 67:23–43, 1994) claimed that the symplectic design on 64 points is the only SDP design on 64 points admitting an abelian regular automorphism group (an abelian difference set). We show in this paper that there is an SDP design on 64 points that is not isomorphic to the symplectic design and yet admits the group C 8 × C 4 × C 2 as a regular automorphism group. This abelian difference set is the first in an infinite family of abelian difference sets whose designs have the SDP and yet are not isomorphic to the symplectic designs of the same order. We define a new method for establishing the non-isomorphism of the two families. [ABSTRACT FROM AUTHOR]

Quasi difference sets are introduced as a tool to produce partial linear spaces. We characterize geometry and automorphisms of configurations decomposable into components induced by quasi difference sets. In particular, we are interested in series of cyclically inscribed copies of a fixed configuration. [ABSTRACT FROM AUTHOR]

There are two construction methods of designs from $(n,m)$ -bent functions, known as translation and addition designs. In this article we analyze, which equivalence relation for Boolean bent functions, i.e. $(n,1)$ -bent functions, and vectorial bent functions, i.e. $(n,m)$ -bent functions with $2\le m\le n/2$ , is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial $(n,m)$ -bent functions and isomorphism of addition designs are the same concepts for all even $n$ and $m\le n/2$. Further, we show that extended-affine inequivalent Boolean bent functions in $n$ variables, whose translation designs are isomorphic, exist for all $n\ge 6$. This implies, that isomorphism of translation designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence. However, we do not observe the same phenomenon for vectorial bent functions in a small number of variables. We classify and enumerate all vectorial bent functions in six variables and show, that in contrast to the Boolean case, one cannot exhibit isomorphic translation designs from extended-affine inequivalent vectorial $(6,m)$ -bent functions with $m\in \{ 2,3 \}$. [ABSTRACT FROM AUTHOR]

We introduce almost supplementary difference sets (ASDS). For odd m, certain ASDS in ℤ m that have amicable incidence matrices are equivalent to quaternary sequences of odd length m with optimal autocorrelation. As one consequence, if 2m − 1 is a prime power, or m ≡ 1 mod 4 is prime, then ASDS of this kind exist. We also explore connections to optimal binary sequences and group cohomology. [ABSTRACT FROM AUTHOR]

Difference sets are subsets of a group satisfying certain combinatorial property with respect to the group operation. They can be characterized using an equality in the group ring of the corresponding group. In this paper, we exploit the special structure of the group ring of an Abelian group to establish a one-to one correspondence of the class of difference sets with specific parameters in that group with the set of all complex solutions of a specified system of polynomial equations. The correspondence also develops some tests for a Boolean function to be a bent function. [ABSTRACT FROM AUTHOR]

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An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF's regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies. [ABSTRACT FROM AUTHOR]

Wilson (1972) proved that for fixed positive integers d and r such that 2 d 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 d th 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. [ABSTRACT FROM AUTHOR]

Since Horadam and de Launey introduced the cocyclic framework on combinatorial designs in the 1990s, it has revealed itself as a powerful technique for looking for (cocyclic) Hadamard matrices. Ten years later, the series of papers by Kotsireas, Koukouvinos and Seberry about Hadamard matrices with one or two circulant cores introduced a different structured approach to the Hadamard conjecture. This paper is built on both strengths, so that Hadamard matrices with cocyclic cores are introduced and studied. They are proved to strictly include usual Hadamard matrices with one and two circulant cores, and therefore provide a wiser uniform approach to a structured Hadamard conjecture.

A bent function from F 2 n to F 2 , n even, can be transformed into a negabent function, or slightly more general into a bent 4 , also called shifted bent function, by adding a certain quadratic term. If n is odd, then negabent functions similarly correspond to semibent functions with some additional property. Whereas bent functions induce relative difference sets in F 2 n × F 2 , negabent functions induce relative difference sets in F 2 n − 1 × Z 4 . We analyse equivalence of negabent functions respectively of their relative difference sets. We show that equivalent bent functions can correspond to inequivalent negabent functions, hence one can obtain inequivalent relative difference sets in F 2 n − 1 × Z 4 with EA-equivalence. We also show that this is not the case when n is odd. Finally we analyse the class of semibent functions that corresponds to negabent functions and show that though partially bent semibent functions always can be shifted to negabent or bent 4 functions, there are many semibent functions which do not correspond to negabent and bent 4 functions. [ABSTRACT FROM AUTHOR]

Generalized perfect binary arrays (GPBAs) were used by Jedwab to construct perfect binary arrays. A non-trivial GPBA can exist only if its energy is 2 or a multiple of 4. This paper introduces generalized optimal binary arrays (GOBAs) with even energy not divisible by 4, as analogs of GPBAs. We give a procedure to construct GOBAs based on a characterization of the arrays in terms of 2-cocycles. As a further application, we determine negaperiodic Golay pairs arising from generalized optimal binary sequences of small length. [ABSTRACT FROM AUTHOR]

Almost designs (t-adesigns) were proposed and discussed by Ding as a certain generalization of combinatorial designs related to almost difference sets. Unlike t-designs, it is not clear whether t-adesigns need also be (t - 1) -designs or (t - 1) -adesigns. In this paper we discuss a particular class of 3-adesigns, i.e., 3-adesigns coming from certain strongly regular graphs and tournaments, and find that these are also 2-designs. We construct several classes of these, and discuss some of the restrictions on the parameters of such a class. We also construct several new classes of 2-adesigns, and discuss some of their properties as well. [ABSTRACT FROM AUTHOR]

We give two new constructions of almost difference sets. The first is a generic construction of (q 2 (q + 1) , q (q 2 - 1) , q (q 2 - q - 1) , q 2 - 1) almost difference sets in certain groups of order q 2 (q + 1) (q is an odd prime power) having ( F q 2 , +) as a subgroup. This construction yields several infinite families of almost difference sets, many of which occur in nonabelian groups. The second construction yields (4 p , 2 p + 1 , p , p - 1) almost difference sets in dihedral groups of order 4p where p ≡ 3 (mod 4) is a prime. Moreover, it turns out that some of the infinite families of almost difference sets obtained produce Cayley graphs which are Ramanujan graphs. [ABSTRACT FROM AUTHOR]

We study finite groups G having a non-trivial, proper subgroup H and D ⊂ G \ H , D ∩ D - 1 = ∅ , such that the multiset { x y - 1 : x , y ∈ D } has every non-identity element occur the same number of times (such a D is called a difference set). We show that | G | = | H | 2 , and that | D ∩ H g | = | H | / 2 for all g ∉ H . We show that H is contained in every normal subgroup of index 2, and other properties. We give a 2-parameter family of examples of such groups. We show that such groups have Schur rings with four principal sets, and that, further, these difference sets determine DRADs. [ABSTRACT FROM AUTHOR]

Rendezvous in the blind fashion is known as one of the most robust and effective ways for cognitive users (CUs) to exchange control information under the dynamic interference of primary users (PUs). Earlier, no existing scheme could guarantee CUs to rendezvous blindly in each timeslot, which makes them experience relatively long exchange delay. Based on the mathematical concept of relaxed cyclic difference set (DS), this paper proposes two novel channel hopping (CH) schemes for multi-radio CUs with and without a synchronized time clock to rendezvous blindly in every timeslot. This achievement of minimum rendezvous delay is especially useful for such delay-sensitive applications as real-time voice and emergency communications. Moreover, based on a special class of DS, the proposed two schemes can closely approach the theoretical lower bound on the number of radio transceivers for CUs to rendezvous blindly in every timeslot and, hence, minimizes their required hardware complexity to exchange control information at the minimum rendezvous delay. Numerical simulation shows that, under dynamic PU interference, the proposed schemes outperform the existing ones in terms of the average throughput and delay of CUs. [ABSTRACT FROM AUTHOR]

A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2‐groups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a linking system of difference sets in non‐2‐groups. We make a connection to Kerdock and bent sets, which provides large linking systems of difference sets in elementary abelian 2‐groups. We give a new construction for linking systems of difference sets in 2‐groups, taking advantage of a previously unrecognized connection with group difference matrices. This construction simplifies and extends prior results, producing larger linking systems than before in certain 2‐groups, new linking systems in other 2‐groups for which no system was previously known, and the first known examples in nonabelian groups. [ABSTRACT FROM AUTHOR]

Asynchronous channel hopping (ACH) systems are widely used for the blind rendezvous among secondary users in cognitive radio networks without requirement for global synchronization and common control channels. We can view an ACH system with n channels as a set of sequences of a common period t on an alphabet of size n satisfying certain rotation closure properties. For any two distinct sequences u, v in an ACH system S, every l ∈{0,1,⋯, t − 1} and any letter j in the alphabet, if there always exists i such that the i-th entries of u and Ll(v) are identical to j where Ll(v) denotes the cyclic shift of v by l, then we say that S is a complete ACH system. Such a system can guarantee rendezvous between any two secondary users who share at least one common channel. It is well known that t⩾n2. When the equality holds, S is called a perfect ACH system. By applying characters of cyclic groups, we obtain a new upper bound on the number of sequences in a perfect ACH system. Furthermore, when q is a prime power and n = q − 1, we present a construction of homogeneous complete ACH systems of period t = 2n2 + o(n2). [ABSTRACT FROM AUTHOR]

A generalized strong external difference family (briefly (v,m;k1,⋯,km;λ1,⋯,λm)-GSEDF) was introduced by Paterson and Stinson in 2016. In this paper, we give some nonexistence results for GSEDFs. In particular, we prove that a (v,3;k1,k2,k3;λ1,λ2,λ3)-GSEDF does not exist when k1+k2+k31, t>1 and v≡t≡1(mod2). Then we use it to obtain some new GSEDFs for m=2. In particular, for any prime power q with q≡1(mod4), we show that there exists a (qt,2;(q-1)2n-1,qt-12;(q-1)2n-2,(q-1)2n-2)-GSEDF, where t=p1p2⋯pn, pi>1, 1≤i≤n, p1,p2,⋯,pn are odd integers. [ABSTRACT FROM AUTHOR]